Differentiation Rules

## Basic Differentiation Rules

Instead of evaluating limits every time, differentiation rules compute derivatives mechanically through algebra.

Rule	Formula	Example
Power rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Constant multiple	$\frac{d}{dx}(c f) = c f'$	Constants factor out
Sum rule	$(f + g)' = f' + g'$	Sum of derivatives

Standard derivatives round out the toolkit: $\frac{d}{dx} e^x = e^x$ , $\frac{d}{dx} \ln x = \frac{1}{x}$ , and $\frac{d}{dx} \sin x = \cos x$ .

## Combination & Chain Rules

Combining functions requires precise rules, especially for products and compositions.

Combination Rule	Formula	Key Distinction
Product rule	$(fg)' = f'g + fg'$	Not the product of derivatives
Quotient rule	$(f/g)' = \frac{f'g - fg'}{g^2}$	Note the minus sign & order
Chain rule	$[f(g(x))]' = f'(g(x)) \cdot g'(x)$	Outer times inner derivative

Common pitfall: Forgetting the chain rule inner derivative. $\frac{d}{dx}\sin(x^2) = \cos(x^2) \cdot 2x$ , not just $\cos(x^2)$ . Omitting $g'(x)$ is the #1 mistake.

The quotient rule, derived by leaning on the product rule --- then

## Basic Differentiation Rules

## Combination & Chain Rules

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Calculus