Limit and diffirentiability of a function

[lmedin02]

Homework Statement

For complex numbers $f$ and $g$, and for $1<p<\infty$ we have $\lim_{t\rightarrow 0}\dfrac{|f+tg|^p-|f|^p}{t}=|f|^{p-2}(\bar{f}g+f\bar{g})$; i.e., $|f+tg|^p$ is differentiable.

I would like to show that the above statement is true.

Homework Equations

The Attempt at a Solution

I have try several attempts in the direction of manipulating the convex function $|x|^p$. But no reasonable conclusions yet.

 

[lmedin02]

So, I have made some progress by rewriting the problem and using L'hopitals rule. But I am still off by a factor of $\dfrac{p}{2}$.

rewriting: $|f+tg|^2=f\bar{f}+tf\bar{g}+t\bar{f}g+t^2g\bar{g}$

When I apply L'Hopitals rule I get $\dfrac{p}{2}|f|^{p-2}(\bar{f}g+f\bar{g})$