Proofs of common derivative shortcuts

The derivative of a function f(x) (noted as f'(x)) is defined as:

$$f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}.$$

An alternate definition is:

$$f'(x) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a}.$$

However, because of how tedious it is to use the definition of a derivative every time the derivative of a function is needed, there are shortcuts known as differentiation formulas which reduce the amount of work needed to differentiate functions. Here, we will prove that these shortcuts work.

Proof

 * 1) Let $$y = x^n$$
 * 2) $$\ln y = n \ln x$$ (Take the natural log of both sides)
 * 3) $$\frac{y'}{y} = \frac{n}{x} $$ (Differentiate both sides with respect to x)
 * 4) $$y' = n \frac{y}{x} = n \frac{x^n}{x} = nx^{(n-1)}$$ (solve for y', substitute x^n for y, simplify)

Shortcut 2: if $$f(x) = \sin x $$, or math>f(x) = \cos x then $$f^{(n)}(x)$$...
$$f^{(n)}(x) = \begin{cases} f^{(0)}(x) = f(x), & n\mod 4=0 \\ f^{(1)}(x),       & n\mod 4=1 \\ f^{(2)}(x),       & n\mod 4=2 \\ f^{(3)}(x),       & n\mod 4=3 \end{cases} $$

Proofs of cases coming soon.