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 $\displaystyle\lim_{x \to a} \frac{f(x) - f(a)}{x-a}$.

This works on all terms with a single function $x^n$ in it.

• $\frac{d}{dx} x^n = nx^{x-1}$

The derivative of $e^x$ is $e^x$!

• $\frac{d}{dx} e^x = e^x$ when $k = 1$
• $\frac{d}{dx} e^{kx} = ke^{kx}$ in general

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.

Works on all terms with two functions multiplied together.

* $\frac{d}{dx} f(x)g(x) = f'(x)g(x) + f(x)g'(x)$

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 $g(x)^2$

There isn't much of a way to remember these other than noticing the patterns to help recall them.

• $\frac{d}{dx} \sin x = \cos x$
• $\frac{d}{dx} \tan x = \sec^2 x$
• $\frac{d}{dx} \sec x = \sec x \tan x$

The cofunctions of these are all negative and, well, co-

• $\frac{d}{dx} \cos x = -\sin x$
• $\frac{d}{dx} \cot x = -\csc^2 x$
• $\frac{d}{dx} \sec x = -\cot x \csc x$

Each one of these three also have the same format of derivative which helps you remember.

This rule is for functions nested into eachother¹⁾.

• $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

This is recursive! For example, $f(g(h(x)))$ will chain rule to:

• $\frac{d}{dx} f(g(h(x))) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$

and so on…

Think of it as peeling layers of an onion!

Because the derivative of $e^x$ is $e^x$ a change of base with natural log²⁾ can be used to get the derivative with any logarithm. BlackPenRedPen explains this better lol.

• $\frac{d}{dx} \ln(x) = \frac{1}{x}$ for $x > 0$
• $\frac{d}{dx} a^x = ln(a)a^x$
• $\frac{d}{dx} \log_{a}x = \frac{1}{x\ln a}$ for $x > 0$

Remember that you can't take the $\log$ of a negative number, and always try to break down logs to simpler term sums using log properties to not give yourself a massive-ass headache before differentiating.