====== Derivative Rules ======
These rules are shortcuts to taking the [[math:derivative|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 $e^x$!

  * $\frac{d}{dx} e^x = e^x$ when $k = 1$
  * $\frac{d}{dx} e^{kx} = ke^{kx}$ in general((this is derived from an extension of chain rule, it __only__ works on constants $k$))

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 $g(x)^2$__//**

===== Derivatives of Trig Functions =====
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.

==== Inverse Trig Functions ====
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.

===== Chain Rule =====
This rule is for functions nested into eachother((i.e. $f(g(x))$)).

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

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!

===== Logarithmic Functions =====
Because the derivative of $e^x$ is $e^x$ a change of base with natural log(($\ln$ or $\log_{e}x$)) can be used to get the derivative with any logarithm. [[https://www.youtube.com/watch?v=oBlHiX6vrQY|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 [[math:log_exp|log properties]] to not give yourself a massive-ass headache before differentiating.

===== Inverse Functions =====
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.
  * $\frac{d}{dx} f^{-1} (x) = \frac{1}{f'(f^{-1}(x))}$