Derivative of Energy: 1st Step Solution

[Buddy711]

I want to show that
if the energy is the integral :

<br /> E = \frac{1}{2} \int^{\infty}_{-\infty} u_{t}^2 \ dx<br />

then the derivative of the energy with respect to time t is

<br /> \frac{dE}{dt} = - \int^{\infty}_{-\infty} u_{xt}^2 + f&#039;(u) u_{t}^2 \ dx<br />

What is the first step can you suggest?
Thanks~!


ps.
<br /> u : u(x,t)<br />

 

[HallsofIvy]

Leibniz's formula says that
\frac{d}{dx}\int_{a(x)}^{b(x)} f(x,t) dt= f(x, b(x))- f(x,a(x))+ \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x} dt

I have no idea how you got that "f'(u)" in there since you say nothing about a function, f, before that.