Calc2.12

From Exampleproblems

Find the derivative of $f(x)\,$ with respect to $x\,$ : $f(x)=\int_{0}^{x}e^t\,dt\,$

Here we have an integral as our function. To take the derivative of an integral, we use the Second Fundamental Theorem of Calculus. It says that the derivative of an integral from a constant to $x\,$ of a function $f(t)\,$ is simply $f(x)\,$ . This makes complete sense because the integral and derivative functions are inverses of each other. Taking an integral first and then taking the derivative should give us the same thing we started with. And, the interval $[a,x]\,$ , where $a\,$ is just some constant, is an arbitrary interval based on the value of the variable $x\,$

$f'(x)=e^x\,$