[Math] Prove that a function is differentiable everywhere using the limit definition of a derivative

Question

I have thw function $$f(x) = (e^x+x^4)\cos(-2x) $$ which is defined for all $x \in \mathbb{R}$.

How do I prove it's differentiable for all x using the limit definition? I'm neither getting conclusive results with $$\lim_{h\to 0} \frac{f(x+h) – f(x)}{h}$$ nor with $$\lim_{x\to x_0} \frac{f(x) – f(x_0)}{x-x_0} $$ when I plug in the function for $f(x)$ and try to rearrange anf simplify it.

What am I missing?

Any help would be appreciated, thanks!

Edit: Is the trick here to Show that all 4 'partial' functions are differentiable on $\mathbb{R}$, thus making the composition of them differentiable in $\mathbb{R}$ as well?

Answer 1

HINT: we have $$\frac{f(x)-f(x_o)}{x-x_o}=\frac{(e^x+x^4)\cos(-2x)-(e^{x_0}+x_0^4)\cos(-2x_0)}{x-x_0}$$ and use $$\cos(-x)=\cos(x)$$