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