Skip to main content\(\def\d{\displaystyle }
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Handout Derivative Rules
Rules for specific types of functions.
- Constant functions
\(\displaystyle (k)' = 0\)
- Power functions
\(\displaystyle (x^{n})' = nx^{n-1}\)
- Exponential functions
-
\((a^{x})' = a^{x}\ln(a)\)
- Logarithmic functions
\(\displaystyle (\ln(x))' = \frac{1}{x}\)
- Trigonometric functions
-
\((\sin(x))' = \cos(x)\text{.}\)
\((\cos(x))' = -\sin(x)\text{.}\)
\((\tan(x))' = \sec^{2}(x) = \frac{1}{\cos^{2}(x)}\text{.}\)
\((\arcsin(x))' = \frac{1}{\sqrt{1-x^{2}}}\text{.}\)
\((\arctan(x))' = \frac{1}{1+x^{2}}\text{.}\)
Rules for combinations of functions.
- Constant multiples
\(\displaystyle (k\cdot f(x))' = k\cdot f'(x)\)
- Sum and difference
\(\displaystyle (f(x) \pm g(x))' = f'(x) \pm g'(x)\)
- Products (the product rule)
\(\displaystyle (f(x)\cdot g(x))' = f'(x)g(x) + f(x)g'(x)\)
- Quotients (the quotient rule)
\(\displaystyle \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^{2}}\)
- Compositions (the chain rule)
\(\displaystyle (f(g(x)))' = f'(g(x))\cdot g'(x)\)
