Derivative and Integration Examples Step By Step | CALCULUS Problems for Beginners

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[00:00:00]Today we are going to explore two fundamental calculus tools taking derivatives and finding indefinite integrals. First we will find the derivative of 5x and then we will integrate 15 y^2. To begin let's visualize the function y = 5x so we remember what it looks like. We plot it on an xycoordinate plane and it is a straight line going upward with a constant slope. Now let's find the derivative of 5x. Because 5 is just a constant multiplied by x raised to the first power, the constant can stay outside while we differentiate x 1st. By the power rule, x becomes 1 * x 0. Since x is just 1, the derivative simplifies to 5. Now let's switch gears and look at integration. We want the indefinite integral of 15 y^2 with respect to y. We begin with the integral symbol and use the constant multiple rule. We can bring the 15 out in front of the integral. So we have 15 * the integral of y^2. Then apply the power rule for integrals.

[00:01:05]Increase the exponent by 1 and divide by that new exponent giving yub 3 plus the constant of integration. Finally multip by 1/3 to see that the result is 5 yub plus a single constant. To recap, the derivative of 5x is just 5 since the slope of that straight line is constant and the integral of 15 y^2 gives 5 yub + c. Showing how constants and powers behave in calculus.