Calc2.14

From Exampleproblems

Find the derivative of $f(x)\,$ with respect to $x\,$ : $f(x)=\int_{4}^{x^2}\sin (e^t)\,dt\,$

This problem uses the Second Fundamental Theorem of Calculus but it also uses the Chain Rule. Since the upper limit of the integral is not plain $x\,$ , then we must multiply by the derivative of the inside function just as we would in any other chain rule problem.

$f'(x)=\sin (e^{x^2})(2x)\,$

The $\sin (e^{x^2})\,$ came from plugging the upper limit $x^2\,$ into the inner function, $f(t)\,$ and the $2x\,$ came from multiplying by the derivative of $x^2\,$ .