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| math:derivative_rules [2021/09/15 18:59] – created epix | math: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. | ||
| - | * $\frac{d}{dx} x^n = nx^{x-1}$ | + | |
| ===== e^x ===== | ===== e^x ===== | ||
| The derivative of $e^x$ is $e^x$! | The derivative of $e^x$ is $e^x$! | ||
| - | * $\frac{d}{dx} e^x = e^x$ | + | |
| + | * $\frac{d}{dx} e^{kx} = ke^{kx}$ 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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| **// | **// | ||
| **// | **// | ||
| + | |||
| + | ===== 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' | ||
| + | |||
| ===== 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' | + | |
| 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' | + | |
| 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:// | ||
| + | |||
| + | * $\frac{d}{dx} \ln(x) = \frac{1}{x}$ 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: | ||
| + | |||
| + | ===== 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' | ||
