Derivative of an Nasty Exponential Function

[Drakkith]

Homework Statement

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.

Homework Equations

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.

 

[fresh_42]

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.

 

[stevendaryl]

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.

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 [itex]P(s)[/itex].
For example,

• Try [itex]P(s) = 1[/itex]: What is [itex]\int_0^x P(s) ds[/itex]? What's its derivative?
  
• Try [itex]P(s) = s[/itex]: What is [itex]\int_0^x P(s) ds[/itex]? What's its derivative?
• Try [itex]P(s) = s^2[/itex].

Do you see the pattern?

 

[Mark44]

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)##.

 

[Mark44]

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.

 

[Drakkith]

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 [itex]P(s)[/itex].
For example,

• Try [itex]P(s) = 1[/itex]: What is [itex]\int_0^x P(s) ds[/itex]? What's its derivative?
  
• Try [itex]P(s) = s[/itex]: What is [itex]\int_0^x P(s) ds[/itex]? What's its derivative?
• Try [itex]P(s) = s^2[/itex].

Do you see the pattern?

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).

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)##.

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) ##

 

[Mark44]

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) ##

Looks good.

 

[Drakkith]

Thanks all!