is continuously differentiable, and satisfies

for all

and for all

.

Let

denote the gradient of f at x and

the dot product. Prove that

for all

.

I first considered the n=1 case: Fixing

, define

0,\infty)\to \mathbb{R}" alt="g

0,\infty)\to \mathbb{R}" /> by

. We have

for all

. Differentiating with respect to

gives

.

On the other hand,

.

*I do not know how to justify this step however: f is defined on R-0, so how does one go about differentiating with respect to a variable that has a restricted domain?*