Question

If \[f(x)=\int\limits_{-8}^{6\sin(x)}(\cos(t^4)+t)dt, then f'(x)=??\]

Answer 1

Use fundamental thm of calculus
\[f(x) = F(6\sin x) - F(-8)\]
where F(t) is anti derivative of cos(t^4)+t
then take derivative with respect to x (use chain rule)
\[f'(x) = F'(6\sin x)*6\cos x - F'(-8)*0\]
\[= (\cos[(6\sin x)^4]+6\sin x)*6\cos x\]