Derivative of Energy: 1st Step Solution

[Buddy711]

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

$$E = \frac{1}{2} \int^{\infty}_{-\infty} u_{t}^2 \ dx$$

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

$$\frac{dE}{dt} = - \int^{\infty}_{-\infty} u_{xt}^2 + f'(u) u_{t}^2 \ dx$$

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


ps.
$$u : u(x,t)$$

 

[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.