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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.

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