Basic Maths for NEET 2027 | Differentiation from Scratch | Complete Foundation Lecture

[00:00:00]Hello all of you. So, let us discuss about differentiation topic right now in physics particularly, right?

[00:00:11]We have basic mathematics very important point.

[00:00:19]So, we all know that basic maths is very very important. It covers in multiple chapters. Let us in kinematics we use it in in all the chapters we will learn new new physical quantities and in every physical quantity whenever we are discussing, there will be use of mathematics, right? So, in basic maths particularly calculus part is very important.

[00:00:47]The calculus part is very very important. And [snorts] today's lecture is about differentiation as I already told you, but calculus part we are actually dealing in differential calculus as well as integral calculus.

[00:01:08]Differential calculus as well as integral calculus.

[00:01:12]In kinematics let us say we learn about displacement, velocity, acceleration.

[00:01:17]So, we use differentiation there, we use integration there. In [snorts] all the chapter let us in gravitation, electrostatics, or every chapter at least one or two questions minimum we will have of basic mathematics there. Without this particular topics, it is very difficult.

[00:01:39]So, that is why here we are going to discuss particularly exclusively on calculus part. Okay? Let's go ahead. Let us say I am talking about functions.

[00:01:53]Before starting about differentiation, let us understand the functions part, sir.

[00:02:00]What is a function? Any relation between variables any relation between variables, we will call it as a function. Like for example, if I want to take uh y is equal to x + 2x + 5.

[00:02:23]I have taken a equation, sir.

[00:02:25]Y is a variable, x is a variable, and y is equal to 2x + 5 is a relation between these variables. And how this equation or let us say very frequently we will use, let us say we know uh area of a circle is equal to πr².

[00:02:50]So, this is a formula or a relation that we know. Here, area is equal to πr² for a circle, where r is the radius. So, if I put the radius value 1, the area of the circle is going to become π m² or π units. And if I put radius as 2, what is the area is going to become, sir?

[00:03:15]4π. Yes or no?

[00:03:18]If I put radius as 3, my area is going to become 3 squared, that is 9π. And so on.

[00:03:27]So, you can see if you change the variable r, the variable a is also changing. So, a is a function of r. We can write this as we can write this as a is a function of r. Any relation between variable is called a function, and we have certain types of functions.

[00:03:54]What are the types of function? Let us say algebra function.

[00:04:02]Where we discuss algebra, like example, y = 2x + 3 or y = x² + x. Generally, any function that we are dealing, we will if we have to plot a graph, we will put in y axis as well as x axis. Like let us say you have you are watching a cricket match.

[00:04:26]So, the number of overs bowled will be on the x axis, the number of runs scored will be on the y axis.

[00:04:33]And the whole match the whole match can be represented in one picture. So, graph is a pictorial representation of data.

[00:04:43]So, similarly, whenever we want to tell so much data, like a particle started from point A, went to point B with certain speed, then with certain acceleration, then he changed and he from 10:00 till 11:00, the whole journey, we have a data form. Now, [snorts] that whole journey, if I want to write in one paper, I can plot a graph. And for plotting a graph, I require the function, how the variables are related.

[00:05:14]Correct? I'll take in y axis a quantity and an x axis another quantity. Let us say in y axis I am saying number of runs scored, and in x axis I am taking >> [snorts] >> the number of overs in a match. So, like that, always I represent x y z. We live in a three-dimensional space. Horizontal axis I'm calling x, vertical axis I'm calling y, and into the plane and out of the plane I'm calling as a Z.

[00:05:48]Correct? So, this is the function Y is equal to f of x and algebraic function. Let us say if I am using trigonometry.

[00:06:07]Y is equal to sin x or Y is equal to cos x or Y is equal to tan x like that. If trigonometry is involved, I am naming that function trigonometric function.

[00:06:19]Similarly, logarithmic functions.

[00:06:29]Y is equal to log x.

[00:06:31]And let us say exponential function.

[00:06:35]Where my variable is in the exponent.

[00:06:43]So, in physics, we require only uh the parts of mathematics. We don't go in deep about mathematics much, but we take help of mathematics. Even if you are learning differential and integration, >> [snorts] >> we are only taking the how much information we require for calculation, only that much only we are taking.

[00:07:08]Is that clear? Similarly, functions in mathematics, for math students, it will be a big chapter. But for us being a physics students, the slide that is present here is more than sufficient.

[00:07:23]Correct?

[00:07:25]What I should know? Function means it is a relation between variables.

[00:07:30]There are different type of functions. In all the cases, left-hand side, LHS, I have taken Y, which is the vertical axis. And RHS I have taken as X.

[00:07:45]So I can represent all these functions on a paper if I want to write, I will write Y is a function of X. This is a way of representation.

[00:07:58]So this much information on functions for us being a physics students is more than sufficient. There are different type of functions, but the left-hand variable is called independent variable.

[00:08:22]And the right-hand variable is called dependent variable.

[00:08:30]Okay? Why because if you can focus here, right-hand side, I will change every point.

[00:08:38]1 2 3 4 5 6 from 0 to plus infinity I can go, minus infinity I can go.

[00:08:46]The right-hand side variable is a independent variable. It is what variable is there? Independent variable.

[00:08:54]But whereas the left-hand side variable LHS, this depends on the function whether it is a algebra function or trigonometry or how how the >> [laughter] >> Excuse me.

[00:09:07]How the variable is changing, accordingly the independent variable will change. Is that clear?

[00:09:15]So that is why left-hand side variable we are calling independent variable.

[00:09:26]And right-hand side variable we are calling Okay? Left-hand side depends on the function, right-hand side will change independently.

[00:09:40]Okay? So, let us go forward. Let us discuss about the next concept, differential calculus.

[00:09:50]I am having differential calculus as well as integral calculus.

[00:10:07]So, I should understand what what does it mean, the basic idea what we are doing in differentiation or integration.

[00:10:15]I'll give a small example, sir. Let us say I want to measure length of a very big rod and I am having a very small scale available.

[00:10:32]So, with this a small scale, I want to measure length of a very big rod.

[00:10:38]So, first question, is it possible uh that I'll measure the length of the rod with this very small scale?

[00:10:47]Second question, if yes, then how?

[00:10:51]You're understanding what I'm telling?

[00:10:53]So, what we will do here is very simple.

[00:10:59]We will break the rod into multiple pieces.

[00:11:05]Multiple markings we will keep. With a small rod, we will say this is length L1.

[00:11:12]This is length L2.

[00:11:14]This is length L3 and so on.

[00:11:18]Right? And in the end, we will say, let us say they are asking what is the total length of the rod?

[00:11:24]What we are going to do? We are going to add the lengths individually and say total length.

[00:11:31]So, similar to this in mathematics, I'm having a very big function.

[00:11:38]Let us say Y is equal to X squared plus X cubed plus 3 X squared minus 1.

[00:11:42]Something like that, big function. Now, to measure that function from point A to point B, from 0 to 10 or 10 to 20, what is the measurement? So, I have very less time. I can't substitute each value like I have done in the previous slide, like here let us say for this function, each value I have put and then I have measuring the average or measuring the instantaneous values. So, that much time we would not be having. So, for that, what we will do, sir?

[00:12:14]We will use the help of the calculus method.

[00:12:18]First, we will do differentiation. What is differentiation? Whatever the function you are having, break it into the pieces, make the smallest piece possible.

[00:12:29]Make the smallest piece possible.

[00:12:35]All right? So, we will split the function into the smallest piece.

[00:12:43]Then, in the end what we have done, we have added everything. We have joined all the things. So, in integration, we will join.

[00:12:56]So, differentiation is very much resembling with splitting a function.

[00:13:00]Your input will be a function and you will break it into smallest piece.

[00:13:05]That is your output will happen in differentiation. But, integration, it is reverse of differentiation. Your input will be a smallest piece and you will join all the small small pieces and make the value of the total unknown thing. So, that we will do in the integration. So, both of them are reverse to each other. Here input function, output is smallest piece.

[00:13:30]Here, input is the smallest piece, output is the value of the all pieces combined. That is an integration. I hope it is clear. We are good to go to next one, sir.

[00:13:41]So, why we have discussed these points here? Because similar thing we are going to do in the upcoming topic.

[00:13:52]Okay. Right. Let us say, I have put a heading differentiation.

[00:14:03]As we are physics students, first of all, we are going to learn the application part of differentiation in physics, instead of the derivation part.

[00:14:15]The derivation part you will be discussing or you will be knowing in your mathematics class, right? Right now, whatever there is differentiation topic and important concepts that we are learning them and trying to apply them with physics. Let us say, velocity is rate of change of position. So, we have to differentiate position with respect to time. So, that application we are learning. Understood? Okay. So, before going to the formula part of differentiation, let us understand the terminology that we will use.

[00:14:50]For a function, y is equal to f of x.

[00:15:01]Y is the independent variable dependent variable, x is the independent variable.

[00:15:07]RHS variable is independent. LHS variable is dependent. So, for this function, let us say, dx whenever I write with the variable the letter d, small d, dx is smallest element of variable x.

[00:15:36]And dy is smallest element of variable y.

[00:15:44]So, what do you mean by that?

[00:15:48]For example, I am having a mm scale.

[00:16:01]So, this is 1 cm scale. 1 cm is divided into 10 divisions.

[00:16:07]10 divisions is 1 cm. What is the smallest division? What is one division equal to?

[00:16:16]Yes.

[00:16:18]0.1 cm or 1 mm.

[00:16:23]So, the smallest least count of a regular cm scale is 1 mm. That means you can accurately measure 1 mm, the smallest possible length. Like that, if if length is your variable, you are talking about a physical quantity called length.

[00:16:43]So, the length of the pen can be 10 cm or another small pencil 8 cm or another pencil 6 cm. So, length is a variable, let us say. Then, the smallest element of this measurement, that is dx. So, in this case, the dx least count would be 1 mm. So, if if variable x we are talking, then dx is the smallest element. I'll give another example.

[00:17:11]>> [snorts] >> Let us say this is in terms of length.

[00:17:14]Let us say in terms of time, if we see, you have a watch.

[00:17:19]Correct?

[00:17:21]Let us say wall clock I have taken. So, the smallest measurement the wall clock can do is 1 second. The least count of a wall clock is 1 second.

[00:17:33]You're talking about analog clock or digital clock, let us say. It may be 0.1 second. If it is a stopwatch, then 0.001 second.

[00:17:43]Right? So, basically, if time is your then the smallest element is dt. So, dt would be 1 second, like that. I hope it is clear. Next.

[00:17:55]If Y is your variable, then if you put a small d small d in front of this letter, we are talking about the smallest element in that variable. Then, dy by dx, that is small change in independent dependent variable divided by small change in independent variable.

[00:18:20]dy by dx is called derivative of the function.

[00:18:35]Example.

[00:18:39]If velocity is 3t squared plus 2t, then who is the variable in left-hand side?

[00:18:49]Velocity. So, small change in velocity by Who is the smallest variable in right-hand side? Time. So, small change in time is derivative of this function.

[00:19:06]Who is this function? V is equal to f of t.

[00:19:11]You're understanding my point? So, on paper whenever they are writing >> [snorts] >> small change in a variable by small change in another variable, that means LHS is the dependent variable and RHS is the independent variable.

[00:19:28]Understood? Let us say in this in this particular slide, area is equal to pi r squared. So, what we can write? Area is a function of Area is a function of radius. And small change in area by small change in radius is called a derivative of this function.

[00:20:06]Is it clear?

[00:20:09]Right.

[00:20:10]>> [clears throat] >> Next, let us go to next slide, sir.

[00:20:23]So, till now, we have just done introduction only. That if I am having a function, small change in dependent by small change in independent is called derivative of the function. Right? And dy by dx is also written as y' is equal to f' of x.

[00:20:54]So, derivative Derivative is also written as what, sir?

[00:21:07]Derivative is also written as y' or f'(x). Whenever they are saying a dash, that also they are saying small change in dependent by small change in independent. One example I'll tell you, sir.

[00:21:20]Okay?

[00:21:21]Example I'm telling you, and this is not a useful example. Useless example, but just for our understanding purpose, we are going to discuss in the next slide we will do the actual method. This is just for understanding purpose. In this method we will never solve any question.

[00:21:38]Disclaimer I'm giving you in the beginning only. Let us say y is equal to 2x + 3.

[00:21:47]And at x is equal to 1, how much is y?

[00:21:55]5.

[00:21:56]If you take x value as 1, y value as 5.

[00:21:59]Now now what I'm going to do, put x value x1 + dx.

[00:22:12]And if you are changing the independent variable, then dependent variable also will change.

[00:22:20]It will also become y1 + dy. Why?

[00:22:23]Because if you change independent with the smallest element, then whatever the change will be there in the function, that is the smallest for independent and dependent variable, correct?

[00:22:36]Here if you are doing the smallest change, so [snorts] whatever it is visible, that will be the smallest change for the variable y.

[00:22:45]So y will become y1 + dy. So let us substitute, sir.

[00:22:51]So therefore, y1 + dy is equal to 2 into x1 + dx + 3 So, how much is y1?

[00:23:06]5 So, 5 + dy is equal to 2 into x1. What is x1? 1 + 2 into dx + 3. So, 2 + 3 5 and 5 minus 5 0. So, what is remaining sir here?

[00:23:26]dy is equal to two times dx. I hope the mathematics is clear sir.

[00:23:34]2 1s are 2 3 2 + 3 5 5 will come this side 5 minus 5 0. So, the remaining is 2 is equal to 2 dy. Therefore, dy by dx is equal to 2.

[00:23:50]So I already told you in this method we are not going to solve anything. Just for our understanding purpose, if you change x by a smallest element, then the dependent variable also will change because it is depending on that function.

[00:24:06]And the smallest change in dependent variable by smallest change in independent variable is called derivative of the function y is equal to 2x + 3.

[00:24:23]Understood?

[00:24:24]This is the actual meaning.

[00:24:26]Like 2x + 3 is the big rod we were talking. Then the smallest element small change in y by small change in x for this question would be 2.

[00:24:39]Is that clear?

[00:24:40]Right.

[00:24:41]So, now in [clears throat] mathematics there are some derivations based on that derivations we have come up with formulas of differentiation, sir.

[00:25:08]In physics, we are only focusing on learning the formulas and applying them in the different ones, right?

[00:25:17]So, let us directly learn the formulas, sir. If y is equal to x power n, then dy / dx is equal to n into x power n minus 1.

[00:25:43]So, we are again I'm telling you, sir, we are focusing only on the application part.

[00:25:48]How we have to apply this.

[00:25:52]If your question looks like function y is depending on x power something. Like if I want to give an example, uh y is equal to x squared or y is equal to x cubed. The n is a constant here.

[00:26:12]Okay? The n is a constant.

[00:26:15]Now, what is the answer or what is the derivative for this particular function?

[00:26:20]Multiply the power and subtract one from the power. We are doing two things. We are multiplying the power and we are subtracting the power. So, if you focus here, what is the derivative for this one function, sir?

[00:26:34]dy / dx is equal to multiply the power, that is 2 into x power 2 minus 1. n into x power n minus 1.

[00:26:47]So, which will be 2x. So, if your x squared is your function, then 2x will be the smallest element in the function.

[00:26:55]I hope it is clear.

[00:26:57]Yes?

[00:26:59]Next.

[00:27:00]Take your time and solve the next one, sir. Y is equal to x cubed.

[00:27:08]Good.

[00:27:09]What is the answer for this?

[00:27:13]>> [clears throat] >> dy by dx should be equal to multiply the power and subtract one from the power.

[00:27:24]That is 3x squared.

[00:27:28]Understood? Let us Let us do a couple of more examples on this particular function.

[00:27:33]Uh let us say our question looks like y is equal to 1 by x.

[00:27:45]y is equal to 1 by x. Now, I want to bring this in the form of x power n. So, if I bring this in the form of x power n, what is happening?

[00:27:57]It is becoming the function is becoming x power minus one.

[00:28:03]Then, tell me, sir, what is the derivative? dy by dx is equal to multiply the power and subtract one from the power.

[00:28:16]So, it is n into x power n minus one.

[00:28:21]So, which is minus one into x power minus two. And in the numerator, if it is power is negative, that means in the denominator it is x squared.

[00:28:34]Is it clear?

[00:28:35]Similarly, similarly, I can also give you one more example, let us say.

[00:28:42]One more example.

[00:28:46]y is equal root X.

[00:28:50]You want to try?

[00:28:52]Okay.

[00:28:54]Pause it and try, sir.

[00:28:56]Pause the video and you can try.

[00:28:59]Let me continue.

[00:29:00]So, Y is equal to root can be written as power 1 by 2. Now, how do do the derivative? Multiply the power, that is 1 by 2 into X power N minus 1, that is power minus 1.

[00:29:19]So, 1 by 2 into X power 1/2 minus 1 is minus 1/2. And in the numerator, if it is negative, you can write denominator positive.

[00:29:31]And if the power is 1 by 2, the power is 1 by 2, it is nothing but root.

[00:29:39]So, if the function is X squared, we will multiply the power and subtract one from the power. So, 2 into X power 2 minus 1. If it is X cubed, then we'll multiply the power and subtract one from the power. That is 3 X squared. This is a very important formula.

[00:30:03]In physics, how it is coming, sir? We know that let us say, velocity is in in the question they have given, sir.

[00:30:14]Let us say, velocity is differentiation of position.

[00:30:19]And they told position of a particle depending on the time, 2 T squared plus 3 T. Now, they are asking velocity. So, what we will do? We will differentiate the velocity.

[00:30:33]Understood? With the help of these formulas.

[00:30:36]Got it? So, that is the reason we should focus more on differentiation. And I already told you integration is what? It is exactly reverse of differentiation.

[00:30:50]So, once you learn differentiation properly, then integration also will become very handy.

[00:30:57]Correct? Right. Next, second formula.

[00:31:02]If y is equal to x So, what is happening now, please see, sir.

[00:31:17]y is equal to x So, if nothing is there in the power, nothing is there in the power, that means how much is the value of n here, sir?

[00:31:29]What is the value of n? 1 Correct? So, what we will do? dy by dx is equal to multiply the power 1 into x power 1 minus 1.

[00:31:44]So, something power zero is always equal to what, sir?

[00:31:49]1 So, 1 into 1 is 1. So, if y is equal to x, then differentiation derivative is how much, sir? 1 Please copy quickly, sir.

[00:32:01]Yes.

[00:32:14]Next.

[00:32:15]Next, try to understand, sir, the third formula. If y is equal to constant They have given us y is equal to constant.

[00:32:27]Okay?

[00:32:30]They have given us y is equal to constant. If I give example, y is equal to 12, y is equal to 13, or y is equal to five, pure number. So, if y is equal to a constant, then what is a derivative? Derivative is zero.

[00:32:46]Why? Because we can write here it is nothing but 12 into one. Nothing is there, no variable. There is no uh independent variable.

[00:32:58]So, we can write 12 into x power zero.

[00:33:02]And if if the power is zero, we will after derivative we will multiply n into x power one. So, zero into zero minus one.

[00:33:12]Anything multiplied by zero is zero. So, the derivative of this function, so something power zero means what we will do, sir?

[00:33:22]Yes.

[00:33:23]Zero into something, that is zero.

[00:33:26]So, three formulas in algebra I've told you, sir. If x power n, then multiply the power and subtract one from the power for derivative. If y is equal to x, directly proportional, then derivative is one.

[00:33:41]If y is equal to constant, then derivative is zero.

[00:33:45]These three are the formulas, important.

[00:33:47]There are more formulas. I'll try to discuss in the next slide.

[00:33:51]I hope you have copied here.

[00:33:53]Yes.

[00:33:54]I can change the slide. Thank you.

[00:34:02]So, fourth formula.

[00:34:10]If y is equal to u plus v.

[00:34:20]Let us say they have given y is equal to x squared plus root x.

[00:34:29]Something plus something, where u is also function of X, V is also function of X. Then, the derivative What the derivative will become?

[00:34:41]dy / dx is equal to If you apply derivative on the LHS and RHS, it will become derivative of U plus derivative of V.

[00:34:54]Okay? That means we have to differentiate X squared, we have to differentiate root X, and add both the derivative. This is the formula for A plus U plus V function.

[00:35:05]So, dy / dx is equal to X squared. So, power is two, sir. Multiply the power and subtract one from the power. X squared derivative is 2x. In the last slide also, we have seen.

[00:35:20]Plus root X. I hope you remember in the last slide, what was the root X derivative? We will multiply the power and subtract one from the power. 1 / 2 root X will come.

[00:35:31]How much it comes, sir? If Y is equal to root X, then derivative is 1 / 2 root X.

[00:35:43]It's derivative You can also call as Y dash.

[00:35:46]So, if Y is equal to root X, Y dash is 1 / 2 root X.

[00:35:51]Multiply the power and subtract one from the power. So, what is the derivative here?

[00:35:57]Directly, we can skip the calculation.

[00:36:00]I'll write 1 / 2 root X.

[00:36:04]So, anything in the form of U plus V, differentiate U and differentiate V.

[00:36:11]More examples we will discuss, sir. Let us go to the next formula.

[00:36:17]If Y is equal to U into V, so this is a multiplication rule. This rule. How these formulas are derived that anyway in mathematics you will discuss sir. Right now in physics we only understand the application part. We have to remember the formula then we have to apply in the kinematics or gravitation or electrostatics, right?

[00:36:48]So, let us focus on the formula now. So, differentiation of Y is given as U into V dash plus V into U dash.

[00:37:01]V U dash plus U V dash.

[00:37:06]Is it clear, sir?

[00:37:17]So, this is the formula for multiplication rule. If we discuss an example, it will be more uh easier, sir.

[00:37:26]Let us see.

[00:37:34]We will discuss more examples of this in the next slide because of the space constraint. Uh if Y is equal to sin X, that is trig now we are jumping from algebra to trigonometric functions.

[00:37:50]And the derivative of the sin X function would be cos X.

[00:37:59]If Y is equal to cos X then the derivative of this function would be minus sin X.

[00:38:11]If Y is equal to tan X then the derivative of this function would be secant squared x.

[00:38:23]Okay? Please copy these formulas also, sir. Then we'll go to the next slide.

[00:38:33]Right.

[00:38:34]Let us discuss an example for u into v now.

[00:38:46]u v dash plus v u dash What example I am giving you?

[00:38:53]I am giving you x squared into sine x.

[00:38:57]x squared is a function algebra function, let us say. Sine x is a trigonometric function. Both are multiplied. x squared into sine x is only y.

[00:39:08]Right? So, here u is being multiplied by v.

[00:39:12]So, how to solve this question?

[00:39:15]We will take u value as x squared and v value as sine x.

[00:39:24]Then what is u dash?

[00:39:28]What is the derivative of x squared?

[00:39:30]2x Correct, sir? Multiply the power and subtract one from the power. So, x squared derivative is 2x. Next.

[00:39:39]Then what is v dash?

[00:39:42]Derivative of sine x just now we have seen, it is cos x.

[00:39:47]Now what is the answer y dash? That is dy by dx is equal to u v dash. First function into derivative of second function plus v u dash. Second function into derivative of first function.

[00:40:11]Right? So, what is the derivative of the v dash? cos x What is the derivative of u dash?

[00:40:19]2 x So therefore the answer will become x squared cos x plus 2 x sin x u v dash plus v u dash like that. More examples we will discuss it, but I hope this is it.

[00:40:40]Next.

[00:40:42]More examples we'll discuss in the class.

[00:40:44]Currently uh another formula let us focus. If y is equal to a constant multiplied by a function y is equal to constant multiplied by a function What do you mean by that?

[00:41:07]4 x cubed. Here constant is 4 and the function is x cubed. Now, whenever multiplication is there, we have to take it as u v dash plus v u dash, correct? Yes or no? So what we will take?

[00:41:26]Your u we will take as 4 and v we will take as x cubed.

[00:41:35]What is the derivative of a constant?

[00:41:37]Correct, zero.

[00:41:40]Because I we have just discussed if there is no dependent independent variable, the change in dependent variable will be zero. dy will be zero.

[00:41:47]The smallest element is zero, right?

[00:41:49]Then what is the small change in v dash now?

[00:41:54]x cubed derivative is 3 x squared. Multiply the power and subtract one from the power. Now, y dash is u v dash plus v u dash.

[00:42:08]So What is U for? Into V dash, that is 3 X squared.

[00:42:18]Plus V U dash. V is X cubed. And U dash is zero.

[00:42:25]So, the V U dash term will become zero always.

[00:42:31]When always? Whenever the first term is a constant. Whenever the first term is a constant, a constant is multiplied by a function, then we can differentiate U into V dash.

[00:42:49]We can do U into V dash, but V into U dash, derivative of the constant term, anyway it is going to become zero.

[00:43:01]Anyway, the V U dash term is going to become zero as the derivative of constant is zero. So, no need to write the second half if the one of the function is constant.

[00:43:15]So, now this becomes a formula.

[00:43:17]What formula? A constant multiplied by a function, keep the constant outside and differentiate the function. U V dash is enough. No need to do V U dash because anyway it is zero.

[00:43:30]Is it clear?

[00:43:32]>> [cough] >> Excuse me.

[00:43:34]Right.

[00:43:37]Next.

[00:43:41]Next we will do, sir.

[00:43:44]Example.

[00:43:46]Y is equal to 12 X squared, let us say.

[00:43:51]Then, just now I told you, dy by dx is equal to 12 is outside, derivative of X squared is 2 X. So, the answer will become 24 X. No need to do This is UV dash.

[00:44:08]No need to do VU dash because anyway it is going to become zero. That is what I'm trying to tell you.

[00:44:15]A constant multiplied by a function is U into V dash enough. VU dash anyway zero because X squared will keep outside.

[00:44:24]Derivative of 12 is zero.

[00:44:26]Right.

[00:44:28]Another example, sir.

[00:44:30]Y is equal to 3X squared + 2X + 5. Now something plus something plus something. Now let us differentiate it, sir.

[00:44:45]3X squared. Three is a constant. Keep constant outside.

[00:44:50]Differentiate X squared. What is the derivative of X squared? 2X. That's it.

[00:44:55]No need to write VU dash. Time waste.

[00:44:57]Anyway zero. Plus plus because it is in the form of U plus V. Now again, two is a constant. Keep outside. Differentiate X. Derivative of X power one is how much, sir? One. We have already discussed X squared not 2X. X not one.

[00:45:13]Correct? And root X not one by two root X. And one by X not minus one by X squared. Always you have to subtract one from the power.

[00:45:23]Right. Plus constant X power zero means zero. So derivative of five is zero. So the answer for this question would be 6X + 2.

[00:45:37]Is that clear?

[00:45:39]Next.

[00:45:40]Couple of more formulas are pending, sir. That also we'll discuss now.

[00:45:45]Let us say.

[00:45:50]Next formula.

[00:45:52]Y is equal to U by V.

[00:45:59]That is numerator by denominator. It's a fraction. We have seen addition subtraction. We have seen uh the multiplication u into v. Now we are seeing the division one. Then What is the formula for the fraction? dy by dx is equal to 1 by v squared into v u dash minus u v dash.

[00:46:25]1 by square of denominator into denominator multiplied by derivative of numerator minus numerator multiplied by derivative of denominator. This is the formula for this question. We must keep it in mind. 1 by v squared into v u dash minus u v dash. One more example we will take, sir.

[00:46:48]y is equal to Let us say I've taken tan x.

[00:46:53]Correct?

[00:46:54]So tan x can be written as what?

[00:46:58]sin x by cos x.

[00:47:01]So who is your numerator here?

[00:47:06]Your numerator is sin x.

[00:47:10]Denominator is cos x.

[00:47:15]What is the derivative of sin x?

[00:47:18]What is the derivative of sin x, sir? We have We have seen cos x.

[00:47:22]What is the derivative of cos x?

[00:47:26]Correct. Minus sin x.

[00:47:29]Now we have to write in this form.

[00:47:31]Therefore, differentiation dy by dx is equal to 1 by square of denominator.

[00:47:40]So 1 by square of denominator, sir.

[00:47:43]cos x whole square.

[00:47:45]Okay?

[00:47:47]Into v u dash. That is what, sir? cos x into derivative of numerator v u dash that is again cos x minus u v dash numerator into derivative of denominator which is sin x into what is the derivative of denominator?

[00:48:12]minus sin x So if you observe carefully and what is 1 by cos can be written as as per trigonometry inverse of cos is secant square secant basically square is there so secant square x into cos into cos cos square x minus into minus plus sin into sin sin square x and from Pythagoras theorem we know sin square x plus cos square x is nothing but one So the derivative of this function dy by dx is how much sir? secant square x If your question is in the form of u by v then the derivative is 1 by square of denominator into denominator multiplied by derivative of numerator minus numerator multiplied by derivative of denominator 1 by v square into v u dash minus u v dash I hope it is clear sir. We'll change next slide.

[00:49:20]next if y is equal to log x then the derivative of log x is 1 by x if y is equal to e power x then derivative of that would be e power x only there is no change in the exponential function question is e power x answer is also e power x.

[00:49:53]Let us Let us look one or two examples of this model also.

[00:49:59]Let us say they have given y is equal to 3x squared into log x.

[00:50:07]So, you can see 3x squared is algebraic function, log x is logarithmic function. It is in the form of u into v.

[00:50:15]It is in the form of what, sir? u into v.

[00:50:18]So, what we can write? u v dash plus v u dash. Now, as you are learning differentiation, we need to take the number of steps we are taking, we have to reduce slowly.

[00:50:36]Right now, when I'm giving you examples, I might have given you a lengthy explanation, but in the exam, we will have less rough space.

[00:50:45]So, we need to improve the number of steps we take.

[00:50:49]So, we know multiplication means keep first function as it is.

[00:50:57]Keep first function as it is.

[00:50:59]Differentiate the second function. So, I'll keep open brackets. Next, keep second function as it is.

[00:51:06]Differentiate the first function.

[00:51:09]So, once I wrote like this, now in the second step, next step, I will do the differentiation. What's the derivative of log x? I'll write 1 by x. What's the derivative of 3x squared? Multiply the powers, uh 3 * 2 is 6 into derivative of x squared, sir, x power 1.

[00:51:26]So, multiply the power, subtract 1 from the power. So, the answer is becoming So, 3x plus 6x log x like that.

[00:51:39]Okay. Next.

[00:51:42]Another example.

[00:51:44]If they're saying y is equal to X e power X.

[00:51:49]X is again algebra, e power X exponential two functions are multiplied.

[00:51:56]Now, what I'll do again dy by dx is equal to I'll say uv' into plus vu'. Keep first function as it is, differentiate the second one, plus keep the second function as it is, differentiate the first one.

[00:52:14]Okay?

[00:52:15]So, what is the derivative of e power X?

[00:52:19]Correct.

[00:52:20]It is visible on the board only. E power X derivative is e power X.

[00:52:25]Plus derivative of X, what is the derivative of X? Multiply the power, subtract one from the power. Derivative of X would be one. So, the answer for this question would be X e power X plus e power X.

[00:52:40]I hope it is clear.

[00:52:42]So, for you, I'll do one more thing, sir.

[00:52:46]I'll do one more thing. I'll try to write whatever we have discussed so far, all formulas in one sheet.

[00:52:55]So, that it will be very, very helpful for you. So, please read and write, sir, whatever we are writing. So, try to understand. So, power X power n, multiply the power, subtract one from the power.

[00:53:11]Next.

[00:53:13]Y is equal to X. What is the derivative?

[00:53:18]Derivative is one.

[00:53:20]Next.

[00:53:22]Y is equal to constant, then derivative is what, sir?

[00:53:28]Zero. Yes or no?

[00:53:31]Next.

[00:53:34]Y is equal to U plus V addition, then y' is u' plus v'.

[00:53:44]Next.

[00:53:46]y is equal to u into v.

[00:53:51]y dash is u v dash plus v u dash.

[00:53:58]Next.

[00:54:01]u by v Then what is y dash?

[00:54:07]1 by v squared into v u dash minus u v dash. Denominator into derivative of numerator minus numerator into derivative of denominator.

[00:54:20]These are all you can say trigonometric relations.

[00:54:24]Next Sorry. Uh algebraic relations formulas.

[00:54:28]Next to trigonometry.

[00:54:31]So if y is equal to sin x, then what is the derivative, sir?

[00:54:35]cos x If y is equal to cos x, what is the derivative?

[00:54:45]minus sin x Correct, sir.

[00:54:47]Good. Next. y is equal to tan x.

[00:54:52]Then its derivative will be secant squared x.

[00:54:56]I'll give you homework, sir. You have to solve y is equal to inverse of sin cosecant x inverse of cos secant x inverse of tan cot x Now I This is your homework, sir.

[00:55:21]You have to find the derivatives of these.

[00:55:25]How you will find also I'll tell you.

[00:55:26]You have to find using u into v rule.

[00:55:31]This rule.

[00:55:33]Is that clear?

[00:55:34]cosecant x we will write 1 by cos Sorry.

[00:55:38]1 by sin. Secant we will write 1 by cos.

[00:55:40]tan we will write 1 by cot. Is that clear?

[00:55:46]Yes?

[00:55:50]Okay.

[00:55:59]cot we will write 1 by tan. So, numerator u we will take as 1, denominator we will take as a function and we'll apply 1 by v squared into v u dash minus u v dash. So, that is your homework, sir.

[00:56:11]Okay? Next.

[00:56:13]We have also seen what?

[00:56:16]Logarithm. y is equal to These are all formulas for trigonometry. We have to keep it in mind. Trigonometry functions and their derivatives. Logarithmic, sorry, algebraic functions and their derivatives. Logarithmic function and its derivative is what, sir? 1 by x.

[00:56:35]Next, e power x.

[00:56:39]Exponential function does not change. e power x derivative will be still e power x. Is that clear?

[00:56:49]So, please copy all these results and try to learn them properly so that we will use it in physics, in kinematics and other chapters.

[00:56:58]Okay? Next.

[00:57:01]I hope up to here everything is fine.

[00:57:12]Next, chain rule of differentiation.

[00:57:27]So, before going to chain rule, I would like to tell you pause the video, try to learn the formulas.

[00:57:35]So, if that is lecture one, this would be called as lecture two.

[00:57:40]Once you learn the differentiation formulas and their application on your own, then this would be very much helpful for you.

[00:57:49]And when we have to use this chain rule, that first I'll give you introduction.

[00:57:54]Let us say, your question is looking like Y is equal to sin 3x squared.

[00:58:03]Or your question is looking like Y is equal to log 4x + 5.

[00:58:12]Something like that.

[00:58:13]Or your question is looking like under root of x squared plus x minus one.

[00:58:22]So, if let us say you have question of only sin x, then you know derivative of sin x is cos x. Or if you know the derivative question is 3x squared, you know the derivative 6x. But algebra function and trigonometry in function inside another function.

[00:58:44]Here logarithm inside another algebra.

[00:58:47]Uh under root algebra inside another algebra.

[00:58:50]Root of x you know 1 by 2 root x. X squared you know derivative 2x. But when both of them combine together, then we have to do chain rule of differentiation. This is very important topic, sir. It is like a level two.

[00:59:06]Now, if Y is equal to a function f of x or there is a function inside another function.

[00:59:28]If there is outer function as well as an inner function then assume inner function as a new variable t a new variable t or z or x any like a new variable we have to assume. Whom we have to assume?

[00:59:57]The inner function.

[00:59:59]So then dy by dx can be written that is the derivative of the function of this question dependent variable y and independent variable x. So dy by dx can be called dy by dt into dt by dx. This is the method or how we solve such type of questions. Why because see in mathematics whenever you multiply and divide same quantities nothing change should happen. dy by dx is still equal to dy by dx but we have modified how we write instead of dy by dx we are writing dy by dt into dt by dx. So what difference it will bring let us focus.

[01:00:56]Okay?

[01:00:58]Let us focus with example number one.

[01:01:01]Understood? What is the example? y is equal to sin 3x squared. y is equal to sin 3x squared. So normal differentiation I can't do first thing. So I have to do chain rule. In chain rule what is the step number one?

[01:01:23]Assume inner function, whatever is the inner function as a variable t.

[01:01:32]So, what I'll write here, you can see.

[01:01:36]Inside bracket, we are having 3x ^ 2.

[01:01:39]So, inner function must be t. So, then what will happen? Y Sorry.

[01:01:48]Inner function t is equal to 3x ^ 2.

[01:01:56]Then I'm assuming inner function as t. Then what is the outer function will become?

[01:02:07]sin 3x ^ 2 can be called as t. Now, what was the function was there, function inside function, I wrote inner function separately, outer function separately. In this particular thing, the dependent variable is y and independent variable is t. So, for this function, what is the derivative?

[01:02:32]dy / dt. And I can say sin x derivative is cos x. Like that, sin t derivative will be cos t.

[01:02:43]But, t you have assumed as per the question, what is t?

[01:02:51]3x ^ 2. Good.

[01:02:54]So, we have found now differ- differentiation of outer function. sin of something derivative is cos of something. Now, let us come back to inner function.

[01:03:08]Dependent variable is t.

[01:03:10]Independent variable is x. So, derivative will be dt / dx.

[01:03:15]Constant keep outside. x squared derivative. Yes, 2x.

[01:03:20]So, dt by dx would be how much, sir? 6x.

[01:03:25]So, if 3x squared is called inner function, then 6x is called derivative of inner function. Now, with the help of chain rule, dy by dx, instead of writing directly, I will write dy by dt, and I will write dt by dx, which is nothing but cos 3x squared, which is dy by dt, into dt by dx, which is 6x.

[01:03:58]So, if you look properly, what is the shortcut for this, sir?

[01:04:04]Whenever an outer function and inner function is there, okay? So, the derivative would be dy by dt. If y is equal to sin t, sin of something is outer function, then dy by dt is derivative of Whenever outer function and inner function are there, what we will do, sir?

[01:04:37]We will do differentiate outer function, which is dy by dt. In English, it is called derivative of outer function. And what is dt by dx? If t is called inner function, because we have assumed t as inner function, then dt by dx is derivative of inner function.

[01:04:59]So, we will multiply with derivative of inner function.

[01:05:07]So, whenever there is function inside function, differentiate inner function, differentiate outer function, multiply them both.

[01:05:19]sin 3x ^ 2 sin of something cos of something 3x ^ 2 derivative 6x multiply them directly we can solve. Is that clear?

[01:05:29]We can do the previous question as well.

[01:05:33]You can see log of something.

[01:05:36]log of something sir So y is equal to log 4x + 5 So obvious we will take inner function as a t log t. So what is the derivative of log t? 1 by t.

[01:05:56]So derivative of outer function log of something 1 by something log t 1 by t into inner function 4x + 5 4 outside derivative of x is 1 + constant derivative 0.

[01:06:13]So 4x + 5 derivative is 4 this is derivative of inner function and log of something is 1 by something. So the answer for this question would be very simpler terms derivative of outer function into derivative of inner function.

[01:06:29]Similarly third question you can do as a homework.

[01:06:33]Is that clear?

[01:06:34]So we have discussed now What is What do you mean by functions?

[01:06:38]Function is y is equal to f of x LHS variable is LHS variable is We have learned about functions.

[01:06:54]Any relation between variable is called a function.

[01:06:57]Uh left hand side is called dependent right hand side is called independent and it can be written as y is equal to f of x.

[01:07:04]or LHS is equal to f of RHS variable.

[01:07:08]And types, what are the types?

[01:07:12]Trigonometry, logarithmic, exponential, algebra. This much information is enough.

[01:07:19]In functions, coming to calculus, big rod, if we break into small piece, differentiation.

[01:07:27]All small piece, if we join, integration, that we are going to do in the next video.

[01:07:35]Next class, next lecture. Now, after that we have seen the terminology.

[01:07:41]If x is a variable, dx is nothing but smallest element, smallest change. dy, smallest change. dy by dx is called derivative.

[01:07:51]Correct?

[01:07:53]The terminology we have learned.

[01:07:55]Then we went to the formulas.

[01:07:57]We have certain 14 formulas we have given, some formulas for algebra, some formulas for trigonometry, logarithmic, exponential.

[01:08:05]By using those formulas, we have to learn the application part in differentiation, that we are going to apply in uh kinematics. Let us say the next chapter would be kinematics. So, we will say displacement, velocity, acceleration.

[01:08:22]The physical quantities that we learn, and how these are related, the velocity is nothing but rate of change of position.

[01:08:32]If we have displacement, and we differentiate, we will get velocity.

[01:08:37]Similarly, acceleration, rate of change of velocity is called acceleration.

[01:08:45]So, if velocity is given, and acceleration is required, we will differentiate.

[01:08:51]And I told you reverse of differentiation is going to be integration. So, you have acceleration, and you want velocity, we will integrate. Similarly, reverse of differentiation is integration. You have velocity and you want displacement. We will integrate. So, calculus is very major part in kinematics as well as other chapters.

[01:09:15]Okay? I hope you have liked the explanation. Wherever you are getting doubt, please pause it, rewind it and go through it again.

[01:09:25]Thank you all of you.