Calculus Question Pls Help ASAP !!

Question

d/dx ∫x^2,2 e^x^3 dx

Find the derivative with respect to x of the integral from 2 to x squared of e raised to the x cubed power, dx.

Answer 1

First we have a definite integral going from a constant to a function of x. If we can rewrite the problem to go from a constant to just x, then we can use the first fundamental theorem of calculus to find an answer.

f(x)= (2, x²) ∫e^{x^3}dx

g(x) = (2,x) ∫e^{x^3}dx (same except the bounds are now from a constant to x which allows the first fundamental theorem to be used)

Defining a similar function were the upper bound is just x then allows us to say f(x)=g(x²) which allows us to say that f'(x)=g'(x²)=g'(x²)*2x (by the chain rule) and g(x) is written so that we can easily take its derivative using the fundamental theorem which says the derivative of an integral from a constant to x is equal the the inside of the integral

g'(x) = e^{x^3}

g'(x²)=e^{(x^2)^3}=e^{x^6}

so we know f'(x) = g'(x²)*2x

f'(x) = 2xe^{x^6}