What is the derivative of the integral of x^2t^2 from 0 to x?

[Appa]

Homework Statement

d/dx (\int^{x}_{0} x²t²dt)
So the problem is to solve the derivative of the integral \int x²t²dt from 0 to x.

Homework Equations

d/dx (\int^{x}_{a} f(t)dt) = f(x)

The Attempt at a Solution

I'm really unsure of how this should be computed but this was my guess:

d/dx (\int^{x}_{0} x²t²dt) = d/dx (1/3x²(x)³ -(1/3x²(0)³)) = d/dx (1/3x⁵) = 5/3x⁴

So, first I calculated the integral with respect to t and then derivated it with respect to x. But it feels wrong. I don't know how to treat the function x²t² because the variable x is both part of the function and an endpoint of the interval for integration.

 

[Dick]

You don't have to treat x^2 as 'part of the function'. It's just a constant multiplying the function t^2. You can take it out of the integral. If you have to do more complicated problems where the x and t are mixed up so you can't do that, check out the Leibniz Integral Rule. So your answer is correct.