TheoremConstant Factor Rule of Differentiation
\begin{equation*} \lzoo{x}{k\fe{f}{x}}=k\lzoo{x}{\fe{f}{x}} \end{equation*}Alternatively, if \(y=\fe{f}{x}\text{,}\) then \(\lzoo{x}{ky}=k\lz{y}{x}\text{.}\)
AppendixBDerivative FormulasΒΆ permalink
\(k\text{,}\) \(a\text{,}\) and \(n\) represent constants; \(u\) and \(y\) represent functions of \(x\text{.}\)
| Basic Formulas | Chain Rule Format | Implicit Derivative Format |
| \(\lzoo{x}{k}=0\) | ||
| \(\lzoo{x}{x^n}=nx^{n-1}\) | \(\lzoo{x}{u^n}=nu^{n-1}\lzoo{x}{u}\) | \(\lzoo{x}{y^n}=ny^{n-1}\lz{y}{x}\) |
| \(\lzoo{x}{\sqrt{x}}=\frac{1}{2\sqrt{x}}\) | \(\lzoo{x}{\sqrt{u}}=\frac{1}{2\sqrt{u}}\lzoo{x}{u}\) | \(\lzoo{x}{\sqrt{y}}=\frac{1}{2\sqrt{y}}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\sin}{x}}=\fe{\cos}{x}\) | \(\lzoo{x}{\fe{\sin}{u}}=\fe{\cos}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\sin}{y}}=\fe{\cos}{y}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\cos}{x}}=-\fe{\sin}{x}\) | \(\lzoo{x}{\fe{\cos}{u}}=-\fe{\sin}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\cos}{y}}=-\fe{\sin}{y}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\tan}{x}}=\fe{\sec^2}{x}\) | \(\lzoo{x}{\fe{\tan}{u}}=\fe{\sec^2}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\tan}{y}}=\fe{\sec^2}{y}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\sec}{x}}=\fe{\sec}{x}\fe{\tan}{x}\) | \(\lzoo{x}{\fe{\sec}{u}}=\fe{\sec}{u}\fe{\tan}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\sec}{y}}=\fe{\sec}{y}\fe{\tan}{y}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\cot}{x}}=-\fe{\csc^2}{x}\) | \(\lzoo{x}{\fe{\cot}{u}}=-\fe{\csc^2}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\cot}{y}}=-\fe{\csc^2}{y}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\csc}{x}}=-\fe{\csc}{x}\fe{\cot}{x}\) | \(\lzoo{x}{\fe{\csc}{u}}=-\fe{\csc}{u}\fe{\cot}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\csc}{y}}=-\fe{\csc}{y}\fe{\cot}{y}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\tan^{-1}}{x}}=\frac{1}{1+x^2}\) | \(\lzoo{x}{\fe{\tan^{-1}}{u}}=\frac{1}{1+u^2}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\tan^{-1}}{y}}=\frac{1}{1+y^2}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\sin^{-1}}{x}}=\frac{1}{\sqrt{1-x^2}}\) | \(\lzoo{x}{\fe{\sin^{-1}}{u}}=\frac{1}{\sqrt{1-u^2}}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\sin^{-1}}{y}}=\frac{1}{\sqrt{1-y^2}}\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\sec^{-1}}{x}}=\frac{1}{\abs{x}\sqrt{x^2-1}}\) | \(\lzoo{x}{\fe{\sec^{-1}}{u}}=\frac{1}{\abs{u}\sqrt{u^2-1}}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\sec^{-1}}{y}}=\frac{1}{\abs{y}\sqrt{y^2-1}}\lz{y}{x}\) |
| \(\lzoo{x}{e^x}=e^x\) | \(\lzoo{x}{e^u}=e^u\lzoo{x}{u}\) | \(\lzoo{x}{e^y}=e^y\lz{y}{x}\) |
| \(\lzoo{x}{a^x}=\fe{\ln}{a}a^x\) | \(\lzoo{x}{a^u}=\fe{\ln}{a}a^u\lzoo{x}{u}\) | \(\lzoo{x}{a^y}=\fe{\ln}{a}a^y\lz{y}{x}\) |
| \(\lzoo{x}{\fe{\ln}{x}}=\frac{1}{x}\) | \(\lzoo{x}{\fe{\ln}{u}}=\frac{1}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\fe{\ln}{y}}=\frac{1}{y}\lz{y}{x}\) |
| \(\lzoo{x}{\abs{x}}=\frac{\abs{x}}{x}\) | \(\lzoo{x}{\abs{u}}=\frac{\abs{u}}{u}\lzoo{x}{u}\) | \(\lzoo{x}{\abs{y}}=\frac{\abs{y}}{y}\lz{y}{x}\) |
TheoremSum/Difference Rule of Differentiation
\begin{equation*} \lzoo{x}{\fe{f}{x}\pm\fe{g}{x}}=\lzoo{x}{\fe{f}{x}}\pm\lzoo{x}{\fe{g}{x}} \end{equation*}TheoremProduct Rule of Differentiation
\begin{equation*} \lzoo{x}{\fe{f}{x}\cdot\fe{g}{x}}=\lzoo{x}{\fe{f}{x}}\cdot\fe{g}{x}+\fe{f}{x}\cdot\lzoo{x}{\fe{g}{x}} \end{equation*}TheoremQuotient Rule of Differentiation
\begin{equation*} \lzoo{x}{\frac{\fe{f}{x}}{\fe{g}{x}}}=\frac{\lzoo{x}{\fe{f}{x}}\cdot\fe{g}{x}-\fe{f}{x}\cdot\lzoo{x}{\fe{g}{x}}}{\left[\fe{g}{x}\right]^2} \end{equation*}TheoremChain Rule of Differentiation
\begin{equation*} \lzoo{x}{\fe{f}{\fe{g}{x}}}=\fe{\fd{f}}{\fe{g}{x}}\cdot\fe{\fd{g}}{x} \end{equation*}Alternatively, if \(u=\fe{g}{x}\text{,}\) then \(\lzoo{x}{\fe{f}{u}}=\fe{\fd{f}}{u}\cdot\lzoo{x}{u}\text{.}\)
Alternatively, if \(y=\fe{f}{u}\text{,}\) where \(u=\fe{g}{x}\text{,}\) then \(\lz{y}{x}=\lz{y}{u}\cdot\lz{u}{x}\text{.}\)