Video Transcript
Find d by d𝑥 of the integral evaluated between five and 𝑥 to the fourth power of the natural log of seven plus 𝑡 cubed with respect to 𝑡.
Now, as tempting as it may be, we absolutely do not want to evaluate the integral here. Instead, we′re going to recall the second fundamental theorem of calculus. This says that if 𝑓 is a continuous function and 𝑎 is a constant, then the integral evaluated between 𝑎 and 𝑥 of 𝑓 of 𝑡 with respect to 𝑡 is equal to capital 𝐹 of 𝑥 minus capital 𝐹 of 𝑎, where capital 𝐹 prime of 𝑥 is equal to the function of 𝑥. And remember, capital 𝐹 is the antiderivative of the function. So how does this help us?
We′re going to differentiate both sides of the equation with respect to 𝑥. And we see that we can separate our derivative into d by d𝑥 of capital 𝐹 of 𝑥 minus d by d𝑥 of capital 𝐹 of 𝑎. Remember, capital 𝐹 of 𝑎 will give us a constant. This means it′s independent of 𝑥. And so, its derivative is going to be zero. So we’re left with the derivative of capital 𝐹 of 𝑥 with respect to 𝑥 or capital 𝐹 prime of 𝑥. Earlier though, we said that capital 𝐹 prime of 𝑥 was equal to 𝑓 of 𝑥. And we therefore see that the derivative with respect to 𝑥 of the integral evaluated between 𝑎 and 𝑥 of 𝑓 of 𝑡 with respect to 𝑡 is equal to 𝑓 of 𝑥.
Let′s see how we can use this to evaluate the derivative with respect to 𝑥 of the integral between five and 𝑥 to the fourth power of the natural log of seven plus 𝑡 cubed with respect to 𝑡. We can say that in our example, 𝑓 of 𝑡 is equal to the natural log of seven plus 𝑡 cubed. And we use the rules we developed earlier. And we see that the derivative with respect to 𝑥 is equal to d by d𝑥 of capital 𝐹 of 𝑥 to the fourth power minus capital 𝐹 of five. We split our derivative up. And then we recall that the derivative of capital 𝐹 of five with respect to 𝑥 is going to be equal to zero, since it′s going to be independent of 𝑥.
And so, we see that the derivative with respect to 𝑥 of the integral evaluated between five and 𝑥 to the fourth power of the natural log of seven plus 𝑡 cubed with respect to 𝑡 is the derivative of capital 𝐹 of 𝑥 to the fourth power with respect to 𝑥. We′re going to need to be really careful here when evaluating d by d𝑥 of capital 𝐹 of 𝑥 to the power of four. We′re going to be differentiating a function of a function, or a composite function. So we′re going to need to use the chain rule.
This says that if 𝑦 is a function in 𝑢 and 𝑢 itself is a function in 𝑥, then the derivative of 𝑦 with respect to 𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥. So this means that the derivative of capital 𝐹 of 𝑥 to the power of four with respect to 𝑥 is equal to the derivative of capital 𝐹 of 𝑥 to the power of four, which is just 𝐹 prime of 𝑥 to the power of four multiplied by the derivative of 𝑥 to the power of four with respect to 𝑥. That’s four 𝑥 cubed.
Now, we said earlier that 𝐹 prime of 𝑥 was equal to 𝑓 of 𝑥. So we can say that 𝐹 prime of 𝑥 to the power of four is equal to 𝑓 of 𝑥 to the power of four. Since we said that 𝑓 of 𝑡 was equal to the natural log of seven plus 𝑡 cubed, we know that 𝑓 of 𝑥 to the fourth power is equal to the natural log of seven plus 𝑥 to the fourth power cubed times four 𝑥 cubed. 𝑥 to the fourth power cubed is 𝑥 to the power of 12. And we can see that the derivative with respect to 𝑥 of the integral evaluated between five and 𝑥 to the fourth power of the natural log of seven plus 𝑡 cubed with respect to 𝑡 is equal to four 𝑥 cubed times the natural log of seven plus 𝑥 to the power of 12.
