Video: Calculating the Derivative of a Definite Integral Where the Upper Limit Is a Function of π‘₯

Calculate d/dπ‘₯ ∫_(0) ^(π‘₯Β³) cos⁴𝑑 d𝑑.

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Video Transcript

Calculate the derivative with respect to π‘₯ of the integral from zero to π‘₯ cubed of cos to the power of four of 𝑑 with respect to 𝑑.

We’re being asked to work out the derivative of an integral with a function of π‘₯ as one of its limits. But this is not as complicated as it looks. In fact, we can use a generalization of the fundamental theorem of calculus. This says that if 𝑒 is a differentiable function of π‘₯ and 𝑓 is continuous, then the derivative with respect to π‘₯ of the integral between π‘Ž and 𝑒 of 𝑓 of 𝑑 with respect to 𝑑 is equal to 𝑓 of 𝑒 times d𝑒 by dπ‘₯. We can apply this directly to our problem since, in our case, 𝑒 is equal to π‘₯ cubed, π‘Ž is equal to zero, and 𝑓 of 𝑑 is equal to cos to the power of four of 𝑑.

In order to work out the derivative of the given integral, we therefore need to find only 𝑓 of 𝑒 and d𝑒 by dπ‘₯. Now, with 𝑒 equal to π‘₯ cubed, d𝑒 by dπ‘₯, by the power rule, is equal to three π‘₯ squared. And with 𝑓 of 𝑑 equal cos to the power four of 𝑑, since 𝑑 is a dummy variable, we have 𝑓 of 𝑒 equals cos to the power of four of 𝑒. And since 𝑒 is equal to π‘₯ cubed, this is equal to cos to the power of four of π‘₯ cubed.

So now, we have the derivative with respect to π‘₯ of the integral between zero and π‘₯ cubed of cos to the power of four of 𝑑 d𝑑 is equal to 𝑓 of 𝑒 d𝑒 by dπ‘₯. Where 𝑓 of 𝑒 is equal to cos to the power of four of π‘₯ cubed, and d𝑒 by dπ‘₯ is three π‘₯ squared. If we rearrange this, we have three π‘₯ squared times cos to the power of four of π‘₯ cubed.

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