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Table of Contents
Derivative Rules
These rules are shortcuts to taking the derivative of a function instead of the long definition with $\displaystyle\lim_{h \to 2} \frac{f(x+h) - f(x)}{h}$ or $\displaystyle\lim_{x \to a} \frac{f(x) - f(a)}{x-a}$.
Power Rule
This works on all terms with a single function $x^n$ in it.
* $\frac{d}{dx} x^n = nx^{x-1}$
e^x
The derivative of $e^x$ is $e^x$!
* $\frac{d}{dx} e^x = e^x$
Be careful not to confuse this rule with power rule, and also to remember to use product or quotient rule if there is another function in the term.
Product Rule
Works on all terms with two functions multiplied together.
* $\frac{d}{dx} f(x)g(x) = f'(x)g(x) + f(x)g'(x)$
Quotient Rule
Works on all terms with two functions in a fraction.
* $\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) + f(x)g'(x)}{g(x)^2}$
DON'T FORGET THE $g(x)^2$ DON'T FORGET THE $g(x)^2$ DON'T FORGET THE $g(x)^2$
Chain Rule
This rule is for functions nested into eachother1)
* $\frac{d}{dx} f(g(x)) = f'(g(x)) + g'(x)$
This is recursive! For example, $f(g(h(x)))$ will chain rule to:
* $\frac{d}{dx} f(g(h(x))) = f'(g(h(x))) + g'(h(x)) + h'(x)$
and so on…
Think of it as peeling layers of an onion!
