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 general¹⁾

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-

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

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

Again, memorization unless you wanna prove each statement during a test or something…

• $\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx} \arctan x = \frac{1}{x^2 + 1}$
• $\frac{d}{dx} \operatorname{arcsec} x = \frac{1}{|x| \sqrt{x^2 - 1}}$

Cofunctions again have the same negative pattern.

• $\frac{d}{dx} \arccos x = \frac{-1}{\sqrt{1-x^2}}$
• $\frac{d}{dx} \operatorname{arccot} x = \frac{-1}{x^2 + 1}$
• $\frac{d}{dx} \operatorname{arccsc} x = \frac{-1}{|x| \sqrt{x^2 - 1}}$

Easiest way I think to remember them is that the sine is basically Pythagorean Theorem sqrt'd, tangent is the only positive one and doesn't have a sqrt, and sec is the complicated one with the absolute value and the $x^2-1$ is swapped. For the cofunctions, I remember with the above negative pattern.

This rule is for functions nested into eachother²⁾.

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

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 log³⁾ 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.

It is probably helpful to have the corresponding $x$ and $y$ value pairs that you are trying to get the inverse derivative of to make your life easier.

• $\frac{d}{dx} f^{-1} (x) = \frac{1}{f'(f^{-1}(x))}$