Calc2.15

From Exampleproblems

$f(x)=\int_{x^2}^{3x^4}\cos (t)\,dt\,$

Again, we use the Second Fundamental Theorem along with the Chain Rule. Most calculus books have very few examples of this type of problem. A general formula for this would be

$\frac{d}{dx}\left[\int_{a(x)}^{b(x)}g(t)\,dt\right]=g(b(x))b'(x)-g(a(x))a'(x)\,$

So, here, we have

$f'(x)=\cos (3x^4)(12x^3)-\cos (x^2)(2x)\,$