# Homework Help: Gradient operation proof

1. Nov 25, 2012

### ohlala191785

1. The problem statement, all variables and given/known data
Show that the operation of taking the gradient of a function has the given property. Assume u and v are differentiable functions of x and y and that a and b are constants.

Operation: (∇(u))n = n*un-1*∇u

2. Relevant equations

The gradient vector of f is <∂f/∂x,∂f/∂y>, where f is a function of x and y (in other words f(x,y)).

3. The attempt at a solution

I tried proof by induction, but I have a lot of gaps.

Base, n=1: ∇u = n*u0∇u. But what is u0 if u is a function?

Assume n = k: ∇uk = n*uk-1∇u

For k=k+1, ∇uk+1 = n*uk∇u.

Isn't proof by induction for sums though? I can't seem to identify the sum here.

Any help would be greatly appreciated!

2. Nov 25, 2012

### Dick

You have got to mean $\nabla (u^n)=n u^{n-1} \nabla u$, $(\nabla(u))^n$ doesn't mean anything. Just use the chain rule for partial derivatives.

3. Dec 1, 2012

### ohlala191785

Oh I see. Thanks!