Question

Find F'(x)
a) F(x)= integral (top bound x, lower bound -4) (t-1)dt

Answer 1

I'm sorry I have no idea how to write integrals in a normal text form

Answer 2

\[\int\limits_{x}^{-4} -12x^2(4x^3 -1)dx\]

Answer 3

You want this:

\[\frac{ d }{ dx } \int\limits_{-4}^{x}(t-1)dt\]

This is one of those fundamental theorem of calculus things. In general:

\[\frac{ d }{ dx }\int\limits_{g(x)}^{f(x)}h(t)dt = h(f(x))\cdot f'(x) - h(g(x)) \cdot g'(x)\]

Applying this idea gives:

\[\frac{ d }{ dx }\int\limits_{-4}^{x}(t-1)dt = (x-1)(1) - (-4-1)(0) = x-1\]