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Table of Derivatives

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    Differentiation Rules

    Sum Rule

    \(\displaystyle \frac {d} {dx} (\textcolor{blue}{f(x)} + \textcolor{brown}{g (x)}) = (\textcolor{blue}{f(x)} + \textcolor{brown}{g (x)}) '= \textcolor{blue}{f'(x)} + \textcolor{brown}{g'(x)} \)
    Constant Multiple Rule \( \displaystyle \frac {d} {dx} (c\textcolor{blue}{f(x)}) = (c\textcolor{blue}{f(x)})'= c\textcolor{blue}{f'(x)} \)
    Product Rule \( \displaystyle \frac {d} {dx} (\textcolor{blue}{f(x)}\textcolor{brown}{g(x)}) = (\textcolor{blue}{f(x)}\textcolor{brown}{g(x)})'= \textcolor{blue}{f'(x)}\textcolor{brown}{g(x)} + \textcolor{blue}{f(x)}\textcolor{brown}{g'(x)} \)
    \( \displaystyle \frac {d} {dx} (\displaystyle \frac {1} {\textcolor{blue}{f(x)}}) = -\displaystyle \frac {\textcolor{blue}{f'(x)}} {(\textcolor{blue}{f(x)})^2} \)
    Quotient Rule \( \displaystyle \frac {d} {dx} \left(\displaystyle \frac {\textcolor{blue}{f(x)}} {\textcolor{brown}{g(x)}} \right) =\left(\displaystyle \frac {\textcolor{blue}{f(x)}} {\textcolor{brown}{g(x)}}\right)'=\displaystyle \frac {\textcolor{brown}{g(x)}\textcolor{blue}{f'(x)} - \textcolor{blue}{f(x)}\textcolor{brown}{g'(x)}} {(\textcolor{brown}{g(x)})^2} \)
    Chain Rule \( \displaystyle \frac {d} {dx} \textcolor{blue}{f(}\textcolor{ brown }{g(x)}\textcolor{blue}{)}= \left(\textcolor{blue}{f(}\textcolor{ brown }{g(x)}\textcolor{blue}{)} \right)'= \textcolor{blue}{f'(}\textcolor{ brown }{g(x)}\textcolor{blue}{)}\textcolor{ brown }{g'(x)} \)

    Derivatives for Elementary Trancendental Functions

    \( \displaystyle \frac {d} {dx} \textcolor{orange}{x}^\textcolor{magenta}{n} = \textcolor{magenta}{n}\textcolor{orange}{x}^{\textcolor{magenta}{n}-1} \)
    \( \displaystyle \frac {d} {dx} e^ \textcolor{orange}{x} = e^ \textcolor{orange}{x} \)
    \( \displaystyle \frac {d} {dx} \textcolor{magenta}{b}^\textcolor{orange}{x} = \textcolor{magenta}{b}^\textcolor{orange}{x}ln(\textcolor{magenta}{b}) \), where \( \, \textcolor{magenta}{b} > 0\)
    \( \displaystyle \frac {d} {dx} \ln(|\textcolor{orange}{x}|) = \displaystyle \frac {1} {\textcolor{orange}{x}} \),\( x \ne 0 \)
    \( \displaystyle \frac {d} {dx} \log_\textcolor{magenta}{b}(|\textcolor{orange}{x}|) = \displaystyle \frac {1} {\textcolor{orange}{x} \, \ln(\textcolor{magenta}{b})} \), \( x \ne 0 \)
    \( \displaystyle \frac {d} {dx} \sin(\textcolor{orange}{x}) = \cos(\textcolor{orange}{x}) \)
    \( \displaystyle \frac {d} {dx} \cos(\textcolor{orange}{x}) = -\sin(\textcolor{orange}{x}) \)
    \( \displaystyle \frac {d} {dx} \tan(\textcolor{orange}{x}) = \sec^2(\textcolor{orange}{x}) \)
    \( \displaystyle \frac {d} {dx} \sec(\textcolor{orange}{x}) = \sec(\textcolor{orange}{x})tan(\textcolor{orange}{x}) \)
    \( \displaystyle \frac {d} {dx} \csc(\textcolor{orange}{x}) = -\csc(\textcolor{orange}{x})cot(\textcolor{orange}{x}) \)
    \( \displaystyle \frac {d} {dx} \cot(\textcolor{orange}{x}) = -\csc^2(\textcolor{orange}{x}) \)
    \( \displaystyle \frac {d} {dx} \sin^{-1}(\textcolor{orange}{x}) = \displaystyle \frac {1} {\sqrt{1-\textcolor{orange}{x}^2}} \)
    \( \displaystyle \frac {d} {dx} \tan^{-1}(\textcolor{orange}{x}) = \displaystyle \frac {1} {1+\textcolor{orange}{x}^2} \)
    \( \displaystyle \frac {d} {dx} \sec^{-1}(\textcolor{orange}{x})= \displaystyle \frac {1} { |\textcolor{orange}{x}| \,\sqrt{\textcolor{orange}{x}^2-1}} \)
    \( \displaystyle \frac {d} {dx} \cos^{-1}(\textcolor{orange}{x}) =- \displaystyle \frac {1} {\sqrt{1-\textcolor{orange}{x}^2}} \)
    \( \displaystyle \frac {d} {dx} \cot^{-1}(\textcolor{orange}{x})=- \displaystyle \frac {1} {1+\textcolor{orange}{x}^2} \)
    \( \displaystyle \frac {d} {dx} \csc^{-1}(\textcolor{orange}{x}) = - \displaystyle \frac {1} { |\textcolor{orange}{x}| \,\sqrt{\textcolor{orange}{x}^2-1}} \)
    \( \displaystyle \frac {d} {dx} |\textcolor{orange}{x}| = sgn(\textcolor{orange}{x}) = \displaystyle \frac {\textcolor{orange}{x}} {|\textcolor{orange}{x}|} , x \ne 0\)

    Table of Derivatives is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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