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math:derivative_rules [2021/09/15 19:16] – Added trig derivatives epixmath:derivative_rules [2022/01/15 02:25] (current) – Added inverse function epix
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 This works on all terms with a single function $x^n$ in it. This works on all terms with a single function $x^n$ in it.
  
-* $\frac{d}{dx} x^n = nx^{x-1}$+  * $\frac{d}{dx} x^n = nx^{x-1}$
  
 ===== e^x ===== ===== e^x =====
 The derivative of $e^x$ is $e^x$! The derivative of $e^x$ is $e^x$!
  
-* $\frac{d}{dx} e^x = e^x$+  * $\frac{d}{dx} e^x = e^x$ 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. 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.
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   * $\frac{d}{dx} \sin x = \cos x$   * $\frac{d}{dx} \sin x = \cos x$
   * $\frac{d}{dx} \tan x = \sec^2 x$   * $\frac{d}{dx} \tan x = \sec^2 x$
-  * $\frac{d}{dx} \sec x = \sec \tan x$+  * $\frac{d}{dx} \sec x = \sec \tan x$
  
 The cofunctions of these are all negative and, well, co- The cofunctions of these are all negative and, well, co-
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   * $\frac{d}{dx} \cos x = -\sin x$   * $\frac{d}{dx} \cos x = -\sin x$
   * $\frac{d}{dx} \cot x = -\csc^2 x$   * $\frac{d}{dx} \cot x = -\csc^2 x$
-  * $\frac{d}{dx} \sec x = -\cot \csc x$+  * $\frac{d}{dx} \sec x = -\csc x \cot x$
  
 Each one of these three also have the same format of derivative which helps you remember. 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 ===== ===== Chain Rule =====
-This rule is for functions nested into eachother((i.e. $f(g(x))$))+This rule is for functions nested into eachother((i.e. $f(g(x))$)).
  
-  * $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$+  * $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
  
 This is recursive! For example, $f(g(h(x)))$ will chain rule to: This is recursive! For example, $f(g(h(x)))$ will chain rule to:
  
-  * $\frac{d}{dx} f(g(h(x))) = f'(g(h(x))) g'(h(x)) h'(x)$+  * $\frac{d}{dx} f(g(h(x))) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$
  
 and so on... and so on...
  
 Think of it as peeling layers of an onion! 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.
 +  * $\frac{d}{dx} f^{-1} (x) = \frac{1}{f'(f^{-1}(x))}$

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