Let be a continuous function on satisfying

Find the value of for all positive real if .

First, we use the fundamental theorem of calculus to differentiate both sides of the given equation with respect to ,

Then, evaluating this at we have

Next, we differentiate this with respect to ,

Finally, since we have

Therefore,

How do you know f(x) is differentiable?

First fundamental theorem of Calculus

Ignore me, thought they were talking about the integral of f(x)

Because of theorem 4.1. Namely, f is a quotient of differentiable functions.