How to compute value of this derivative at x=0

Question

This might sound like a pretty basic thing but I cannot figure this out.

\begin{align} q(x)=\frac{f(x)}{g(x)}=&\frac{\sin(x)}{x}\\ \frac{dq(x)}{dx} = &\frac{x\cos(x)-\sin(x)}{x^2} \end{align}

But what is the value of $\frac{dq(x)}{dx}$ at $x=0$?

I did plot at wolframalpha.com but dont know how to make an argument that goes with it.

Separate functions

Composite function

Answer 1

Any even function $f$ that is differentiable at $x=0$ must have $f'(0) = 0$.

Answer 2

To evaluate the expression

$$\frac{x\cos(x)-\sin(x)}{x^2}$$

you can either use L'Hôpital's rule or the Taylor's expansion of cosine and sine at $x=0$. These give you $$\frac{x\left(1+O(x^2)\right)-x+o(x^2)}{x^2} = \frac{o(x^2)}{x^2}=0$$