Video Transcript
Calculate the derivative with respect to 𝑥 of the integral from zero to 𝑥 cubed of cos to the power of four of 𝑡 with respect to 𝑡.
We’re being asked to work out the derivative of an integral with a function of 𝑥 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 𝑢 is a differentiable function of 𝑥 and 𝑓 is continuous, then the derivative with respect to 𝑥 of the integral between 𝑎 and 𝑢 of 𝑓 of 𝑡 with respect to 𝑡 is equal to 𝑓 of 𝑢 times d𝑢 by d𝑥. We can apply this directly to our problem since, in our case, 𝑢 is equal to 𝑥 cubed, 𝑎 is equal to zero, and 𝑓 of 𝑡 is equal to cos to the power of four of 𝑡.
In order to work out the derivative of the given integral, we therefore need to find only 𝑓 of 𝑢 and d𝑢 by d𝑥. Now, with 𝑢 equal to 𝑥 cubed, d𝑢 by d𝑥, by the power rule, is equal to three 𝑥 squared. And with 𝑓 of 𝑡 equal cos to the power four of 𝑡, since 𝑡 is a dummy variable, we have 𝑓 of 𝑢 equals cos to the power of four of 𝑢. And since 𝑢 is equal to 𝑥 cubed, this is equal to cos to the power of four of 𝑥 cubed.
So now, we have the derivative with respect to 𝑥 of the integral between zero and 𝑥 cubed of cos to the power of four of 𝑡 d𝑡 is equal to 𝑓 of 𝑢 d𝑢 by d𝑥. Where 𝑓 of 𝑢 is equal to cos to the power of four of 𝑥 cubed, and d𝑢 by d𝑥 is three 𝑥 squared. If we rearrange this, we have three 𝑥 squared times cos to the power of four of 𝑥 cubed.
