Calc Topic 3 3 Derivatives of Inverse Functions

Hinzugefügt: 2026-05-06

[00:00:06]okay today we're going to talk about topic 3.1 differentiating inverse functions so first i'd like to review inverse notation because the better you understand inverse notation the easier this topic will be for you this is not a major topic in ap calculus you may see one question show up on the actual ap exam so let's go ahead and review let's start with review let's say if we have a function f of x then the function notation for the coordinate is x comma f of x that's a point on the graph that is on the graph of f okay if we have its inverse let's call it f inverse of x that's the inverse equation for f of x that's the notation for the inverse function of f x then the coordinate would be f inverse of x that's the notation for the y value on the inverse function this represents the y value this x is the x value on the inverse function okay so that is on the graph of the inverse of f well what do you know about inverses what do we know about inverses well if a b is on f of x then b a is on the inverse right so what we will be using is this x value right here represents the x value on the inverse function what does this x value represent on f of x if this is the x value on the inverse then it's the y value on f so this x here will be given to you in a problem they're going to be giving you the x value of a point on the inverse and you have to know that the x value is given here is the x value on the inverse but that's not how you're going to use it you're going to be using it as the y value on f of x okay so the x value on the inverse function is equal to the y value on f of x that's what's going to help you in these calculus problems okay so let's take a look at this particular problem here we have f of x equals x squared and we have the inverse is the square root of x so this point right here is 2 4 so on the inverse it's 4 2 and vice versa it says suppose you were asked to show that the derivative at a point 2 4 is the reciprocal of the derivative of the inverse so let's go ahead and do that let's find the derivative of f of x so if f of x equals x squared and we're dealing with just x is greater than 0 for right now because that's this picture here first quadrant what is the first derivative of f that's just 2x okay i'm gonna make a table i'm gonna call it table a okay they want us to find the derivative at x equals two and the y value would be 4.

[00:05:28]what is the derivative if the x value is 2 what's the derivative of the original function at x equals two and the answer would be four okay so what did you just find looking at this picture you found the instantaneous rate of change of this function at x equals 2. so you found the slope of the tangent line to this point is four okay so we've done that we've done that before let's take a look at the other equation the other equation the inverse function the inverse of f in the first quadrant is the square root of x okay that's the inverse so let's find the derivative of the inverse so this is the inverse this is one notation that represents the derivative of the inverse and we know the square root rule is 1 over 2 square roots of x okay so i'm going to make a table here i'm going to call it table b and we have an x value we have the y value on the inverse and we have the here's another notation the derivative of the inverse function so notice this notation here and notice this notation here they are one in the same okay all right so we're just going to use one point here on the inverse here the original is 2 4 on the inverse the point on the graph is 4 2 and there it is right there what is the derivative what is the instantaneous rate of change at x equals four well let's plug in four you get one over two square roots of four you get one 4 okay did we prove that the derivative of f of x is the reciprocal of the derivative of the inverse at 4 2. so this is the derivative at 2 4 which is 4.

[00:08:19]this is the derivative of the inverse at 4 2 which is 1 4. so compare are they reciprocals of each other yes they are so again here is what you just found one-fourth which is the derivative of the inverse which is what we're going to be finding is equal to 1 over the derivative of the original function f x evaluated at 2.

[00:08:53]so this the reciprocal the derivative of the inverse is the reciprocal of this right here okay or using a different notation was another way to say 2 using this notation over here what represents 2 in this table and the reason why i'm asking you what represents 2 in this table because these are the table values that they're going to be giving you they're going to be giving you a bunch of x's a bunch of f to the the inverse of f of x and the derivative of the inverse of f of x and you'll need to be able to find the information from the given tables okay and you're gonna have to be able to go um i take that back table a is what they're going to give you and you have to know how this information relates back to this table so anyway let's go back to my question the number two in inverse notation inverse function notation is equivalent to what the inverse at x equals four the inverse at x equals four so this notation is the same thing as 2.

[00:10:48]so when you see this given this 4 is the x value on the inverse but how we're going to use it is it's the y value on f of x so again this 4 here right now i'm going to use a different color here let's put it in black note the 4 from this notation is the y value on f of x from table a and table a as you will see in the coming examples is what they will be giving you they will not give you table b okay so you have to know what this means and where to find it so we're used to finding information from a table this one's just kind of backwards okay so if you want to re-watch that to listen to it again and try to get that down if you want to wait to have to reduce some examples that might be a good idea as well okay so this formula basically what i kind of derived in a sense is this right here and this is what you need to memorize and understand the derivative of an inverse is equivalent to the reciprocal of the derivative of the y value on the inverse that goes with this x value this represents the y value on the inverse this represents the x value on the inverse and one more time the y value on the inverse corresponds to the x value in the table a okay or more importantly this x value here corresponds to the y value on f of x look let's take a look at some examples and and let that sink in for a while okay so the first method it would be really nice if we can do this but of course on the ap exam this does not usually happen um but let's say i'm giving you a function and i asked you to find the derivative of its inverse at x equals three well what we could do is find the inverse so a find the inverse of the function so back from um i don't know probably ninth tenth grade algebra one sometime to find an inverse you switch the x's with the y's so our inverse is x equals the square root of y plus five that is the inverse equation okay but now we're going to solve it for y okay we're going to solve it for y so if we square both sides we end up getting y plus five which gives us if we solve for y we get x squared minus five now let's be a little bit more specific we're not talking about any y we're talking about the inverse function equals x squared minus five okay so this is f of x this is the inverse of f of x okay so step b take the derivative of this new equation well isn't that what they're asking us to do find the derivative of the inverse okay so we found the inverse first now we're going to take the derivative of the inverse again i like this notation better it's just easier to write personal preference so what is the derivative the derivative is 2x and then they're going to ask you to evaluate it at x equals 3. again this is x equals 3 on the inverse x equals 3 on the inverse so we're going to go ahead and find the derivative of the inverse at x equals 3.

[00:15:42]okay so the answer is six all right now let's redo this problem using this formula here and let's see if we can get the same answer okay that's on the next page so in order to be able to use this formula we're not going to let's just assume you can't find the inverse because again in calculus most of the problem is you cannot find the inverse so what you first need to do is find the derivative of the equation that's given let's go ahead and find the derivative that's 1 over 2 square roots of u times the derivative of u make sure you understand that is the derivative and the derivative of u is just one okay so just keeping with what is given to us let's not find anything any inverse or anything like that okay the other thing that's given to us x equals three on the inverse function how does that translate to this over here if x equals 3 on the inverse function that means y is equal to 3 on f of x okay so that's what's given to us we didn't have to find anything else well let's just use what's given okay let's look over here let's continue with this can you solve this this we can actually solve if you can't solve it you can use a calculator if it asks you or allows you to use a calculator which we're not going to be using a calculator we're going to be using guess and check because several of them are not calculator problems you have to use guess and check so in this one we're going to go ahead and square both sides and you get x 9 equals x plus 5 therefore x is equal to 4. okay that's based on the given now let's go ahead and evaluate this derivative let's go ahead and evaluate this derivative at x equals 4. so that's one over two square roots of four plus five that's one over that is one over six so what did we just find you just found that what what point did we use what coordinate did we use on f of x well we used four comma what what's the y value 3. so at 4 3 you found the slope of the tangent line or the first derivative is equal to 1 over 6.

[00:19:08]so 4 3 is on f of x and f prime of 4 equals 1 6.

[00:19:26]now how is that going to help us answer the question the question is find the derivative of the inverse well this we we didn't even find the inverse we didn't even use anything with the inverse but what can we conclude now well you can conclude that 3 4 is on the inverse function and the derivative of the inverse oops let's be more specific the derivative of the inverse at x equals three which was our original question to begin with find the derivative of the input inverse of x equals three based on the information what do you think the answer is well we have just shown you previously that they are so your answer is six using this formula and the given information you can find the derivative of the inverse without actually finding the inverse and that's how they're going to want you to do it again you're not going to be able to find the inverse of many equations in calculus so let's go back to the first page i think i forgot to show you what you calculated over here what you calculated in the purple is the slope of the tangent line equals one-fourth that's what we found here slope of this is four slope of this is 1 4.

[00:21:17]okay so what we found on this example is the slope of the tangent line on f of x is 1 6 and at 4 3 so the slope of the tangent line at 3 4 on the inverse is the reciprocal of 1 over 6 which is 6.

[00:21:39]okay so let's take a look at a table problem as we've been used to again the notation another notation is the derivative the derivative of the inverse function at x equals one on the inverse can you find that on the table well they don't give you the inverse they give you g of x so let's think about this this is the x value on the inverse so what does this represent on g of x the y value the y value so what we have learned up to this point is that the derivative of the inverse f at one is equal to the reciprocal of the derivative not at one but at negative one so this is the x value on the inverse of g so that corresponds to the y value so what you do is you find the y value of one on the table and you flip it over so your answer is negative five okay pause the video and try b try it on your own pause the video okay welcome back this is the x value let me use a different color here this is the x value on the inverse well they don't give us the inverse so this is the y value this represents y find that y value right here so the slope of the tangent line is negative 2 at this point on g so what is the slope of the tangent line of the inverse of g oh it's just flipped over and that should be your answer so no work necessary you just have to memorize and understand what this formula means and what it represents okay right let's take a look at method number two which is a method you can use when it's not very easy to solve for y so it says don't solve for y because it's going to be difficult so let's go ahead and follow example number two steps so we are going to switch the x's with the y's to find the inverse we did that back in method number one then we solve for y we're not going to solve for y because that would be nearly impossible it is impossible so what we're going to do instead of solving for y we're going to take the derivative implicitly which we just learned how to do the derivative of both sides with respect to x so the derivative of x is one the derivative of y cubed is three y squared y prime plus y prime implicit differentiation okay step c solve for the derivative so we get y one sorry one equals factor out the y prime and you get three y squared plus one so that yields your y prime equals one over three y squared plus one okay well not any y prime again the inverse derivative is equal to one over three y squared plus y notice when you do this method your derivative is in terms of your y value it's not in terms of your x value and they give you the x value so you can't plug in 10 because this is the derivative in terms of the y value so let's go to step d it says replace the x value on your inverse function this right here from step a above and solve for y okay so this is your inverse right it's telling us that x is equal to 10 on our inverse so i'm going to say okay x equals 10 on our inverse what's the y because we need to know the y well again this could be a calculator problem but this one is not it is possible for you to type those two equations into your calculator and find out where they intersect um that is possible that they will do that on an ap test not very often but it does help happen we are going to do problems where we're just going to use guess and check so guess and check what number can i plug in for y obviously it has to be the same y value that's going to give me a 10 here how about 2 2 cubed plus 2.

[00:28:24]so y is equal to 2.

[00:28:27]10 is given you have to guess and check to find y now you can actually evaluate the derivative of the inverse at x equals 10 is equal to this equation over here this formula and again it's not 3 times ten it's three times your y value squared so that ends up giving us one over thirteen okay so that is one way of doing it but i'm gonna recommend using the formula that i told you to memorize and understand so i've already did an example kind of using this so let's go ahead and let's try this one using this formula here so again stick with what's given okay so f of x is given to us so i would find f prime of x three x squared plus one okay now this number right here is the x value on the inverse function so what is it over here the y value on f of x okay so again we just kind of just did this so we should already know that's guess and check what is x guess and check well x is equal to 2.

[00:30:34]so let's evaluate the derivative at x equals two let's evaluate this derivative at x equals two three over two squared plus one equals thirteen so again the slope of the tangent line is thirteen the first derivative oops let's back up here that's at 2 so what if i asked you what is the derivative of the inverse function at x equals 10 on the inverse what would your answer be well if this is the derivative at y equals 10 on f of x the derivative at x equals 10 on the inverse is the reciprocal and that's what we're trying to get you to understand okay so what are some of the things that we can that we know about f of x and the inverse from this information okay let's start with f of x what point is on f of x well it's 2 10 2 10 on f of x has a tangent line slope of 13.

[00:32:34]the reason why i'm asking you these now is because eventually they're going to ask you to find the equation of the tangent line what else do we know at 10 2 on the inverse function the inverse has a tangent line slope of 1 over 13.

[00:33:12]okay all right let's try you can turn off the video try the next two examples and then come back and check your work if you want to watch one more example we can do that as well or you can do that so again i'm going to be using the information that's given to me okay let me find the derivative of what is given so that's three fourths x squared plus one x equals three is on the inverse function right on the inverse function so y equals 3 on f of x [Music] that's an x so can you guess and check how about this how do we add four four equals one fourth x cubed plus x what do you think x is what number guess just guess and check trial and error try two two cubed is eight eight divided by four is two plus two is four okay so let's find the [Music] derivative at x equals two and three fourths times two squared plus one so the derivative at x equals two on f of x is four so the derivative of the inverse function the derivative of the inverse function at x equals three has a first derivative of one fourth okay i am frozen let's see i didn't get too far so let me um go back and restart um i'm not gonna start here from the very beginning but let me restart my elmo here i know i'm talking too long but i'm trying to take my time so let's let me restart this here you're supposed to be trying this on your own anyway right all right i think we're back back in action so check your work okay what can you tell me well 2 3 is on f of x the slope of the tangent line at 2 3 is 4 3 2 is on the inverse and the slope of the tangent line at 3 2 on the inverse is 1 4.

[00:36:42]all right last one now i would turn off the video and try this one on your own and i'm going to change colors all right check your work if f of x is given i'm going to use its derivative 1 plus cosine of x i'm going to use the x value of pi as my y value did you get the guessing check on this one what would x have to be in order to get pi well i'm assuming this is going to be pi plus zero the sine of pi is zero so yes x equals pi so if i take the derivative of the original at pi the derivative of f of x at pi is one plus a negative one or zero so to answer the question find the derivative of the inverse so find the derivative of the inverse at x equals pi just so happens to be y equals pi so pi pi is on both of them is equal to what that's the question well it's zero flipped over it's undefined okay so what again what can you tell me well that pi pi is on both of the graphs um on f of x you have a horizontal tangent at pi pi pi pi you like that horizontal tangent at pi pi and uh therefore the derivative of the inverse at pi is undefined all right that is the derivative of inverses i know it was a little bit longer video um but i just repeated myself over and over again to try to help this sink in with you if you have to watch it again feel free to watch it again unfortunately it's not easy to explain and it's not a major major part of the actual ap exam but you do need to know it so now you can go try some practice problems