Find the derivative of:
$\displaystyle F(x) = \int_{x^3}^{x^4} (2t-1)^3\, dt$
using the Fundamental Theorem of Calculus
You should show some effort on these problems. We aren't suppose to just give you the answer, we just need to help with your weak points. Here are some properties of integrals that you will need to differentiate the function you were given:
Split the integral using the property:
$\displaystyle \int_a^bf(x)dx=\int_c^bf(x)dx+\int_a^cf(x)dx$
Then you will need to flip the limits of one of the terms so you can differentiate. Use the property:
$\displaystyle \int_a^bf(x)dx=-\int_b^af(x)dx$
You will also need to use the chain rule. Here's an example of how this works.
$\displaystyle g(x)=\int_a^{h(x)}f(t)dt$
$\displaystyle \frac{dg}{dx}=\frac{dh}{dx}\frac{dg}{dh}$
By the FTOC:
$\displaystyle \frac{dg}{dh}=\frac{d}{dh}\int_a^{h(x)}f(t)dt=f(h( x))$
Therefore,
$\displaystyle g'(x)=h'(x)f(h(x))$
Try to use this information to find the derivative. If you still are having trouble, then I will help you further.