# Help computing derivative of a 2 variable integral

• December 4th 2012, 07:09 AM
Maroka93
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
• December 4th 2012, 11:03 AM
TriKri
Re: Help computing derivative of a 2 variable integral
Use the definition of the derivative:

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

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

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

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

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

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

Good luck!
• December 8th 2012, 07:17 AM
TriKri
Re: Help computing derivative of a 2 variable integral
Had any luck solving the problem yet?
• December 8th 2012, 08:07 AM
HallsofIvy
Re: Help computing derivative of a 2 variable integral
I would recommend "Leibniz' rule" which generalizes the "Fundamental Theorem of Calculus":
$\frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt= f(x, u(x))- f(x,v(x))+ \int_{u(x)}^{v(x)}\frac{\partial f}{\partial x}dx$
• December 8th 2012, 12:03 PM
TriKri
Re: Help computing derivative of a 2 variable integral
I think that should be

$\frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = \frac{dv(x)}{dx} f(x,v(x)) - \frac{du(x)}{dx} f(x, u(x)) + \int_{u(x)}^{v(x)}\frac{\partial f(x,t)}{\partial x}dt$
• December 8th 2012, 12:05 PM
HallsofIvy
Re: Help computing derivative of a 2 variable integral
Yes, you are right. I forgot the derivatives of u and v. Thanks.