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Appendix B. Derivative Formulas

\(k\text{,}\) \(a\text{,}\) and \(n\) represent constants; \(u\) and \(y\) represent functions of \(x\text{.}\)
Table B.0.1.
Basic Formulas Chain Rule Format Implicit Derivative Format
\({\lzoo{x}{k} = 0}\)
\({\lzoo{x}{x^n} = nx^{n-1}}\) \({\lzoo{x}{u^n} = nu^{n-1}\lzoo{x}{u}}\) \({\lzoo{x}{y^n} = ny^{n-1}\lz{y}{x}}\)
\({\lzoo{x}{\sqrt{x}} = \frac{1}{2\sqrt{x}}}\) \({\lzoo{x}{\sqrt{u}} = \frac{1}{2\sqrt{u}}\lzoo{x}{u}}\) \({\lzoo{x}{\sqrt{y}} = \frac{1}{2\sqrt{y}}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\sin}{x}} = \fe{\cos}{x}}\) \({\lzoo{x}{\fe{\sin}{u}} = \fe{\cos}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\sin}{y}} = \fe{\cos}{y}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\cos}{x}} = -\fe{\sin}{x}}\) \({\lzoo{x}{\fe{\cos}{u}} = -\fe{\sin}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\cos}{y}} = -\fe{\sin}{y}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\tan}{x}} = \fe{\sec^2}{x}}\) \({\lzoo{x}{\fe{\tan}{u}} = \fe{\sec^2}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\tan}{y}} = \fe{\sec^2}{y}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\sec}{x}} = \fe{\sec}{x}\fe{\tan}{x}}\) \({\lzoo{x}{\fe{\sec}{u}} = \fe{\sec}{u}\fe{\tan}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\sec}{y}} = \fe{\sec}{y}\fe{\tan}{y}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\cot}{x}} = -\fe{\csc^2}{x}}\) \({\lzoo{x}{\fe{\cot}{u}} = -\fe{\csc^2}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\cot}{y}} = -\fe{\csc^2}{y}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\csc}{x}} = -\fe{\csc}{x}\fe{\cot}{x}}\) \({\lzoo{x}{\fe{\csc}{u}} = -\fe{\csc}{u}\fe{\cot}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\csc}{y}} = -\fe{\csc}{y}\fe{\cot}{y}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\tan^{-1}}{x}} = \frac{1}{1 + x^2}}\) \({\lzoo{x}{\fe{\tan^{-1}}{u}} = \frac{1}{1 + u^2}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\tan^{-1}}{y}} = \frac{1}{1 + y^2}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\sin^{-1}}{x}} = \frac{1}{\sqrt{1 - x^2}}}\) \({\lzoo{x}{\fe{\sin^{-1}}{u}} = \frac{1}{\sqrt{1 - u^2}}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\sin^{-1}}{y}} = \frac{1}{\sqrt{1 - y^2}}\lz{y}{x}}\)
\({\lzoo{x}{\fe{\sec^{-1}}{x}} = \frac{1}{\abs{x}\sqrt{x^2 - 1}}}\) \({\lzoo{x}{\fe{\sec^{-1}}{u}} = \frac{1}{\abs{u}\sqrt{u^2 - 1}}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\sec^{-1}}{y}} = \frac{1}{\abs{y}\sqrt{y^2 - 1}}\lz{y}{x}}\)
\({\lzoo{x}{e^x} = e^x}\) \({\lzoo{x}{e^u} = e^u\lzoo{x}{u}}\) \({\lzoo{x}{e^y} = e^y\lz{y}{x}}\)
\({\lzoo{x}{a^x} = \fe{\ln}{a}a^x}\) \({\lzoo{x}{a^u} = \fe{\ln}{a}a^u\lzoo{x}{u}}\) \({\lzoo{x}{a^y} = \fe{\ln}{a}a^y\lz{y}{x}}\)
\({\lzoo{x}{\fe{\ln}{x}} = \frac{1}{x}}\) \({\lzoo{x}{\fe{\ln}{u}} = \frac{1}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\fe{\ln}{y}} = \frac{1}{y}\lz{y}{x}}\)
\({\lzoo{x}{\abs{x}} = \frac{\abs{x}}{x}}\) \({\lzoo{x}{\abs{u}} = \frac{\abs{u}}{u}\lzoo{x}{u}}\) \({\lzoo{x}{\abs{y}} = \frac{\abs{y}}{y}\lz{y}{x}}\)
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