# Derivative of an Nasty Exponential Function

• Drakkith
In summary, the question is asking for the derivative of ##\frac{d}{dx}(exp\int_1^x P(s)\ ds)##, which can be solved using the first part of the Fundamental Theorem of Calculus. The solution involves taking the derivative of the integral and multiplying it by the original function.
Drakkith
Mentor

## 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.

## 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.

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.

mfb and Drakkith
Drakkith said:
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 $P(s)$.
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?

Drakkith and mfb
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)##.

Drakkith
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.

stevendaryl said:
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?

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).

Mark44 said:
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) ##

Drakkith said:
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.

Thanks all!

## 1. What is the derivative of an exponential function?

The derivative of an exponential function is the rate of change of the function at any given point. In other words, it tells us how fast the function is changing at a specific point on the graph.

## 2. How do you find the derivative of an exponential function?

To find the derivative of an exponential function, we use the power rule of differentiation. This means that we multiply the constant in front of the exponential term by the exponent and then subtract 1 from the exponent.

## 3. Can you provide an example of finding the derivative of an exponential function?

Yes, for example, if we have the function f(x) = 3e^x, we would first multiply the constant 3 by the exponent x, giving us 3x. Then, we subtract 1 from the exponent, giving us the final derivative of f'(x) = 3e^x.

## 4. What is the derivative of a nasty exponential function?

The derivative of a nasty exponential function would still follow the same rules as a regular exponential function. However, it may require more steps and simplification due to the complexity of the function. It is important to remember to use the chain rule if the function includes other variables or functions within the exponential term.

## 5. Why is it important to know the derivative of an exponential function?

Knowing the derivative of an exponential function can help us understand the behavior of the function and make predictions about its future values. It is also a fundamental concept in calculus and is used in many real-world applications such as finance, physics, and engineering.

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