Remember that we define the derivative $f'$ of a function $f$ by $$f'(x) = \lim_{h\longrightarrow 0}\frac{f(x+h)-f(x)}{h}.$$ From the definition, one can determine several differentiation rules, three of which are non-intuitive:

The Product Rule: $$\big(f(x)g(x)\big)' = f'(x) g(x) + f(x) g'(x).$$
The Quotient Rule: $$\left(\frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}.$$
The Chain Rule: $$\bigg(f\big(g(x)\big)\bigg)'=f'\big(g(x)\big)g'(x).$$ Below are some examples of applications of these rules.

If $f(x) = 3x^2-4x$ then $f'(x) =$ .
If $f(x) = \sin x - 4 \cos x$ then $f'(x) =$ .
The limit $\lim_{h\longrightarrow 0}\frac{3(2+h)+4-(3\times 2 + 4)}{h}$ defines the derivative of the function $f(x) =$ at the point $x=$ .
If $f(x) = x^2\sin x$ then $f'(x) =$ .
If $f(x) = \frac{\cos x}{x^2}$ then $f'(x) =$ .
If $f(x) = \sin\left(x^2+1\right)$ then $f'(x) =$ .
If $f(x) = \frac{x\tan^2(x)}{1+x^2}$ then $f'(x) =$ .