# Derivative of an Integral problem

• May 16th 2010, 09:56 AM
AlderDragon
Derivative of an Integral problem
I need to find the derivative of an integral, with a limit of integration being a function:

$

g(x) = \int_1^{x^{4}}(t^3+2)dt
$

How do I find g'(x)?
• May 16th 2010, 10:16 AM
CaptainBlack
Quote:

Originally Posted by AlderDragon
I need to find the derivative of an integral, with a limit of integration being a function:

$

g(x) = \int_1^{x^{4}}(t^3+2)dt
$

How do I find g'(x)?

Put:

$h(x)=\int_1^{x}(t^3+2)dt$

Then:

$g(x)=h(x^4)$

and so by the chain rule:

$\frac{d}{dx}g(x)=4x^3 \left. \frac{dh}{dx}\right|_{x^4}$

where the derivative of $h$ is obtained by applying the fundamental theorem of calculus.

CB
• May 16th 2010, 05:49 PM
AlderDragon
Upon closer inspection of my past work I found another method of doing this type of problem. Substitute $x^4$ in for $t$ and multiply by the derivative of $x^4$, $4x^3$:

$

g(x) = \int_1^{x^{4}}(t^3+2)dt
$

$
g'(x) = (x^{12} + 2)4x^3
$

Is this the correct answer? And, if so, is this a different answer than you might obtain using a different method?
• May 16th 2010, 07:42 PM
CaptainBlack
Quote:

Originally Posted by AlderDragon
Upon closer inspection of my past work I found another method of doing this type of problem. Substitute $x^4$ in for $t$ and multiply by the derivative of $x^4$, $4x^3$:

$

g(x) = \int_1^{x^{4}}(t^3+2)dt
$

$
g'(x) = (x^{12} + 2)4x^3
$

Is this the correct answer? And, if so, is this a different answer than you might obtain using a different method?

That is what you get if you finish what I posted in my last post, so yes it is correct. However why you do what you say you do (and what it means for that matter) needs explaining.

CB
• May 17th 2010, 03:18 AM
HallsofIvy
By the way, here is the general Leibniz formula:
$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} F(x,t)dt=$ $F(x, \beta(x))\frac{d\beta(x)}{dx}- F(x,\alpha(t))\frac{d\alpha(x)}{dx}+$ $\int_{\alpha(x)}^{\beta(x)}$ $\frac{\partial F(x,t)}{\partial x} dt$.

It can be proved from the fundamental theorem of calculus and using the chain rule for both the upper and lower limits.