$f(x) = \begin{cases}x^4\left(2 + \sin\frac1x\right) & x\neq 0\\ 0 & x=0.\end{cases}$
Prove that $f$ is differentiable on $\Bbb R$.
I know how to prove that $f$ is differentiable at $x = 0$ by showing that $\lim_{h \to 0} \frac {f(0 + h) - f(0)}h = 0$ and solving it, but how do i prove that it is differentiable if $x$ doesnt equal 0? Is it showing $\lim_{h \to 0} \frac {f(x + h) - f(x)}h?$ Stuck can someone show me how?