# Thread: Help computing derivative of a 2 variable integral

1. ## Help computing derivative of a 2 variable integral

Im not sure if there is a typo, or I'm just not seeing how to solve this:

Compute the derivative w.r.t. t of g(t) := ∫0t f(t,x) dx.

sorry for the iffy notation

2. ## Re: Help computing derivative of a 2 variable integral

Use the definition of the derivative:

$\displaystyle \frac{\partial}{\partial\,t}\int_0^t f(t,\,x)\, \operatorname{d}x \,=\, \lim_{h\to 0} \frac{\int_0^{t+h} f(t+h,\,x)\, \operatorname{d}x - \int_0^t f(t,\,x)\, \operatorname{d}x}{h}$

You see that the first term in the fraction can be written as

$\displaystyle \int_0^{t+h} f(t+h,\,x)\, \operatorname{d}x \,=\, \int_0^t f(t+h,\,x)\, \operatorname{d}x + \int_t^{t+h} f(t+h,\,x)\, \operatorname{d}x$

$\displaystyle =\, \int_0^t f(t+h,\,x)\, \operatorname{d}x + h\,f(t,\,t) + O(h^2)$

$\displaystyle =\, \int_0^t \left(f(t,\,x) + h\,\frac{\partial}{\partial t}f(t,\,x) + O(h^2)\right) \operatorname{d}x + h\,f(t,\,t) + O(h^2)$

From there it should be an easy task to escape the limit formulation and obtain the derivative.

Good luck!

3. ## Re: Help computing derivative of a 2 variable integral

Had any luck solving the problem yet?

4. ## Re: Help computing derivative of a 2 variable integral

I would recommend "Leibniz' rule" which generalizes the "Fundamental Theorem of Calculus":
$\displaystyle \frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt= f(x, u(x))- f(x,v(x))+ \int_{u(x)}^{v(x)}\frac{\partial f}{\partial x}dx$

5. ## Re: Help computing derivative of a 2 variable integral

I think that should be

$\displaystyle \frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = \frac{dv(x)}{dx} f(x,v(x)) - \frac{du(x)}{dx} f(x, u(x)) + \int_{u(x)}^{v(x)}\frac{\partial f(x,t)}{\partial x}dt$

6. ## Re: Help computing derivative of a 2 variable integral

Yes, you are right. I forgot the derivatives of u and v. Thanks.