# Proofs of common derivative shortcuts

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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.

## Shortcut 1: if $f(x) = x^n$ then $f'(x) = nx^{(n-1)}$

#### 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 $f(x) = \cos x$ 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.