Answer to Question #2216 in Calculus for Jess

Question

Find the derivative of the function.
y = int_{cos(x)}^{5 x} cos(u^4) du

Answer 1

lt;img src="/cgi-bin/mimetex.cgi?y%20=%20%5Cint_%7Bf_1%28x%29%7D%5E%7Bf_2%28x%29%7Df%28u%29du%20=%20F%28f_2%28x%29%29%20-%20F%28f_1%28x%29%29%20%5C%5C%20%5Cfrac%7Bdy%7D%7Bdx%7D%20=%20%5Cfrac%7Bdy%7D%7Bdu%7D%20%5Cfrac%7Bdu%7D%7Bdy%7D%20=%20f%28f_2%29f_2%27%28x%29-%20f%28f_1%29f_1%27%28x%29%20%5C%5C%20%5Cfrac%7Bdy%7D%7Bdx%7D%20=%205%5Ccos%28625%20x%5E4%29%20-%20%5Csin%7Bx%7D%5Ccos%28%5Ccos%5E4%7Bx%7D%29" title="y = \int_{f_1(x)}^{f_2(x)}f(u)du = F(f_2(x)) - F(f_1(x)) \\ \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dy} = f(f_2)f_2'(x)- f(f_1)f_1'(x) \\ \frac{dy}{dx} = 5\cos(625 x^4) - \sin{x}\cos(\cos^4{x})">