Question

How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function `h(x)= sqrt(3+r^3) dr` for a=1, `b=x^2`?

Answer 1

I wonder if perhaps you mean to ask how to use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function `h(x)= int_1^(x^2) sqrt(3+r^3) dr`

If so, we'll need the Fundamental Theorem Part 1 and also the chain rule.

`d/(du)(int_1^(u) sqrt(3+r^3) dr) = sqrt(3+u^3)`,

but we've been asked for `d/(dx)(h(x)` so we'll use the chain rule with `u = x^2`

`d/(dx)(int_1^(u) sqrt(3+r^3) dr) = sqrt(3+u^3) * ((du)/(dx)) = 2xsqrt(3+x^6)`,