Let be an everywhere continuous function with

Define another function by

Prove that

Also, compute the values, and .

* Proof. * We start with the equation for and use properties of the integral and the fundamental theorem of calculus to rearrange,

We can pull the out of the integral since it does not depend on (which is what we are taking the integral over). Then, we want to take the derivative with respect to ; however, we need to be careful and use the product rule. Looking closely at the first term we have

Where we used the fundamental theorem to obtain . So, using this method to take the derivative for we have

Now, to evaluate the second and third derivatives at 1 as requested, we first get expressions for and .

Therefore,