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math:derivative_rules [2021/09/20 02:35] – added log functions epixmath:derivative_rules [2022/01/15 02:25] (current) – Added inverse function epix
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   * $\frac{d}{dx} e^x = e^x$ when $k = 1$   * $\frac{d}{dx} e^x = e^x$ when $k = 1$
-  * $\frac{d}{dx} e^{kx} = ke^{kx}$ in general+  * $\frac{d}{dx} e^{kx} = ke^{kx}$ 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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   * $\frac{d}{dx} \cos x = -\sin x$   * $\frac{d}{dx} \cos x = -\sin x$
   * $\frac{d}{dx} \cot x = -\csc^2 x$   * $\frac{d}{dx} \cot x = -\csc^2 x$
-  * $\frac{d}{dx} \sec x = -\cot x \csc x$+  * $\frac{d}{dx} \sec x = -\csc x \cot x$
  
 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 =====
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 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. 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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