How to show $\lim\limits_{t\to 0}\frac{|f+tg|^{p}-|f|^{p}}{t}=pfg|f|^{p-2}$ Could anyone help to compute this limit? Thank you!
This is part of a proof in Analysis by E. Lieb. Let $f$ and $g$ be real numbers and $p>1$, show the following limit:
$$\lim\limits_{t\to 0}\frac{|f+tg|^{p}-|f|^{p}}{t}=pfg|f|^{p-2}$$

The given limit is easily seen to be the (definition of the) derivative of the function $$ t \mapsto |f+gt|^p $$ at $t = 0$. We can calculate this derivative using the chain rule. This comes out to be
$$
p |f+gt|^{p-1} \times \frac{f+gt}{|f+gt|} \times g.
$$
Here, the first factor is the derivative of the function $z \mapsto z^p$ at $|f+gt|$. The second factor is the derivative of the absolute value function at $f+gt$.  Upon substituting $t=0$, we get the claimed result. 
To make the above proof watertight, observe that the absolute value function is differentiable only when the argument is nonzero; so in the above, we have to assume that $f \neq 0$. One would then need to complete the proof by discussing the case $f=0$ separately, as in @Arturo Magidin's answer. 
