A.3: Table of Derivatives
- Page ID
- 92251
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\ket}[1]{\left| #1 \right>}\)
\(\newcommand{\bra}[1]{\left< #1 \right|}\)
\(\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}\)
\(\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}\)
\(\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}\)
\(\newcommand{\op}[1]{\hat{\mathbf{#1}}}\)
\(\newcommand{\expect}[1]{\langle #1 \rangle}\)
\(\newcommand{\dfn}[1]{\emph{\textbf{#1}}}\)
Throughout this table, \(a\) and \(b\) are constants, independent of \(x\text{.}\)
| \(F(x)\) | \(F'(x)=\frac{\mathrm{d}F}{\mathrm{d}x}\) |
| \(af(x)+bg(x)\) | \(af'(x)+bg'(x)\) |
| \(f(x)+g(x)\) | \(f'(x)+g'(x)\) |
| \(f(x)-g(x)\) | \(f'(x)-g'(x)\) |
| \(af(x)\) | \(af'(x)\) |
| \(f(x)g(x)\) | \(f'(x)g(x)+f(x)g'(x)\) |
| \(f(x)g(x)h(x)\) | \(f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)\) |
| \(\frac{f(x)}{g(x)}\) | \(\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\) |
| \(\frac{1}{g(x)}\) | \(-\frac{g'(x)}{g(x)^2}\) |
| \(f\big(g(x)\big)\) | \(f'\big(g(x)\big)g'(x)\) |
| \(F(x)\) | \(F'(x)=\frac{\mathrm{d}F}{\mathrm{d}x}\) |
| \(a\) | \(0\) |
| \(x^a\) | \(ax^{a-1}\) |
| \(g(x)^a\) | \(ag(x)^{a-1}g'(x)\) |
| \(\sin x\) | \(\cos x\) |
| \(\sin g(x)\) | \(g'(x)\cos g(x)\) |
| \(\cos x\) | \(-\sin x\) |
| \(\cos g(x)\) | \(-g'(x)\sin g(x)\) |
| \(\tan x\) | \(\sec^2 x\) |
| \(\csc x\) | \(-\csc x\cot x\) |
| \(\sec x\) | \(\sec x\tan x\) |
| \(\cot x\) | \(-\csc^2 x\) |
| \(e^x\) | \(e^x\) |
| \(e^{g(x)}\) | \(g'(x)e^{g(x)}\) |
| \(a^x\) | \((\ln a)\ a^x\) |
| \(F(x)\) | \(F'(x)=\frac{\mathrm{d}F}{\mathrm{d}x}\) |
| \(\ln x\) | \(\frac{1}{x}\) |
| \(\ln g(x)\) | \(\frac{g'(x)}{g(x)}\) |
| \(\log_a x\) | \(\frac{1}{x\ln a}\) |
| \(\arcsin x\) | \(\frac{1}{\sqrt{1-x^2}}\) |
| \(\arcsin g(x)\) | \(\frac{g'(x)}{\sqrt{1-g(x)^2}}\) |
| \(\arccos x\) | \(-\frac{1}{\sqrt{1-x^2}}\) |
| \(\arctan x\) | \(\frac{1}{1+x^2}\) |
| \(\arctan g(x)\) | \(\frac{g'(x)}{1+g(x)^2}\) |
| \(\textrm{arccsc} x\) | \(-\frac{1}{|x|\sqrt{x^2-1}}\) |
| \(\textrm{arcsec} x\) | \(\frac{1}{|x|\sqrt{x^2-1}}\) |
| \(\textrm{arccot} x\) | \(-\frac{1}{1+x^2}\) |