# Derivative of an Nasty Exponential Function

1. Jan 15, 2017

### Drakkith

Staff Emeritus
1. The problem statement, all variables and given/known data
The question is given just like this:
$\frac{d}{dx}(exp\int_1^x P(s)\ ds)$ = ?
I assume they want me to find the derivative of the whole thing.

2. Relevant equations

3. The attempt at a solution

I'm thinking the first step is:
$\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x P(s)\ ds)(\frac{d}{dx}\int_1^x P(s)\ ds)$

Unfortunately I don't know how to deal with the derivative of the integral now since the X is one of the limits of integration and not part of the function. I'm not sure I've ever seen a problem like this before.

2. Jan 15, 2017

### Staff: Mentor

I've used something similar in one of @micromass ' challenges. I think it helps to write the integral according to the fundamental theorem as $\int_1^x f = F(x) - F(1)$ but I'm not sure.

3. Jan 15, 2017

### stevendaryl

Staff Emeritus
Well, too much of a hint will give away the answer. But to get an idea of how this works, try a couple of examples for $P(s)$.
For example,
• Try $P(s) = 1$: What is $\int_0^x P(s) ds$? What's its derivative?
• Try $P(s) = s$: What is $\int_0^x P(s) ds$? What's its derivative?
• Try $P(s) = s^2$.
Do you see the pattern?

4. Jan 15, 2017

### Staff: Mentor

The first part of the Fundamental Theorem of Calculus says, in essence, that $\frac d{dx} \left(\int_a^x f(t)dt\right) = f(x)$.

5. Jan 15, 2017

### Staff: Mentor

I should add that things have to be exactly as I wrote them. I.e., if we're differentiating with respect to x, x has to be the upper limit of integration. Likewise, if the differentiation is with respect to, say, r, r has to be the upper limit of integration.

Also, although the integral appears to be a definite integral (which would imply that the integral represents a fixed number), it's really a function of x. Different values of x produce different values for the integral.

6. Jan 15, 2017

### Drakkith

Staff Emeritus
Indeed. When I put those values in for P(s) it appears that the derivative to each integral is simply P(s) (or P(x) rather).

So it does. I certainly feel silly now.

So that would mean that: $\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x P(s)\ ds)(\frac{d}{dx}\int_1^x P(s)\ ds) =(exp\int_1^x P(s)\ ds)P(x)$

7. Jan 15, 2017

### Staff: Mentor

Looks good.

8. Jan 15, 2017

### Drakkith

Staff Emeritus
Thanks all!