# Derivative of an Integral

1. Jun 8, 2012

### p75213

1. The problem statement, all variables and given/known data
F is an antiderivative of f, so F’=f.
$$\begin{array}{l} \int_{g(x)}^{h(x)} {f(t)\,dt} = F\left( {h\left( x \right)} \right) - F\left( {g\left( x \right)} \right) \\ \frac{d}{{dx}}\int_{g(x)}^{h(x)} {f(t)\,dt} = F'\left( {h\left( x \right)} \right)h'\left( x \right) - F'\left( {g\left( x \right)} \right)g'\left( x \right) \\ \end{array}$$

2. Relevant equations
Can somebody show how to find the derivative of the following integral?

$$\frac{d}{{ds}}\int_0^\infty {f\left( t \right){e^{ - st}}dt}$$

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jun 8, 2012

### vela

Staff Emeritus
Just reverse the order of integration and differentiation. The fundamental theorem doesn't really apply here.

3. Jun 8, 2012

### Harrisonized

Use integration by parts. Then d/ds every term. Then write the solution as a sum.

Last edited: Jun 8, 2012
4. Jun 8, 2012

### Villyer

I would think the answer is 0, since the definite integral is just a number, and the derivitive of a number if 0.

5. Jun 8, 2012

### Harrisonized

This isn't true. The definite integral for multivariable functions can be a function of a different variable, and if you actually tried the problem, you'd see that the definite integral becomes a function of s.

6. Jun 9, 2012

7. Jun 9, 2012

### p75213

Leibniz's Integral rule as shown by Dick does the trick:

$$\frac{d}{{ds}}\int_0^\infty {f\left( t \right){e^{ - st}}dt} = \int_0^\infty {\frac{\delta }{{\delta s}}f\left( t \right){e^{ - st}}dt} = \int_0^\infty {f\left( t \right)\left( { - t{e^{ - st}}} \right)dt} = \int_0^\infty { - tf\left( t \right){e^{ - st}}dt}$$