$$g(f(x_0))+g'(f(x_0))\cdot f'(x_0) \cdot t + o(t)_{t\to 0} \Rightarrow (g(f(x)))'(x_0)=g'(f(x_0))\cdot f'(x_0) $$
\end{proof}
\begin{cor}
הנגזרת של $x^\alpha $ היא $\alpha x^{\alpha - 1} $, גם אם $\alpha \not\in\mathbb{N} $
\end{cor}
\begin{proof}
$$(x^\alpha)'=(e^{\alpha \ln x})'=(e^{\alpha \ln x})\cdot (\alpha \ln x)' = x^\alpha \cdot \alpha\cdot \frac{1}{x}=\alpha x^{\alpha - 1} $$
\end{proof}
\begin{example}
$$(\cos x)'=\left(\sin (\frac{\pi}{2} - x)\right)'=\cos(\frac{\pi}{2}-x)\cdot (\frac{\pi}{2}-x)' = -\cos(\frac{\pi}{2}-x) = -\sin x $$
$\Leftarrow$
$$(\tan x)'=\left(\frac{\sin x}{\cos x}\right)' = \frac{(\sin x)'\cos x - \sin x (\cos x)'}{\cos^2 x} = \frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}=\frac{1}{\cos^2(x)} $$
\end{example}