Derivative of an Integral problem

**AlderDragon** · May 16th 2010, 09:56 AM

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)?

**CaptainBlack** · May 16th 2010, 10:16 AM

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

**AlderDragon** · May 16th 2010, 05:49 PM

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?

**CaptainBlack** · May 16th 2010, 07:42 PM

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

**HallsofIvy** · May 17th 2010, 03:18 AM

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.

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