Differentiation Rules

Evaluating limits every time is impractical. Differentiation rules make derivative calculation routine algebra.

For combining functions, use these rules:

Product rule: $(fg)' = f'g + fg'$ . Differentiate one factor at a time and add.
Quotient rule: $(f/g)' = \frac{f'g - fg'}{g^2}$ . Note the minus sign and order.
Chain rule: $(f(g(x)))' = f'(g(x))g'(x)$ . Outer times inner.

A product is an area, and the product rule is what its two new

Combine the combination rules with standard derivatives:

Function	Derivative	Function	Derivative
$\sin x$	$\cos x$	$e^x$	$e^x$
$\cos x$	$-\sin x$	$\ln x$	$1/x$

Common Pitfalls:

Product rule error: The derivative of a product is not the product of derivatives. $(fg)' \neq f'g'$ . Always use $f'g + fg'$ .
Chain rule error: For $f(g(x))$ , you must multiply by $g'(x)$ . For example, $\frac{d}{dx}\sin(x^2) = \cos(x^2)\cdot 2x$ , not just $\cos(x^2)$ . Forgetting $g'(x)$ is the #1 mistake.

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