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AppendixBDerivative Formulas

\(k\), \(a\), and \(n\) represent constants; \(u\) and \(y\) represent functions of \(x\).

Basic Formulas Chain Rule Format Implicit Derivative Format
\(\lzoo{x}{k}=0\)
\(\lzoo{x}{x^n}=nx^{n-1}\) \(\lzoo{x}{u^n}=nu^{n-1}\lzoo{x}{u}\) \(\lzoo{x}{y^n}=ny^{n-1}\lz{y}{x}\)
\(\lzoo{x}{\sqrt{x}}=\frac{1}{2\sqrt{x}}\) \(\lzoo{x}{\sqrt{u}}=\frac{1}{2\sqrt{u}}\lzoo{x}{u}\) \(\lzoo{x}{\sqrt{y}}=\frac{1}{2\sqrt{y}}\lz{y}{x}\)
\(\lzoo{x}{\fe{\sin}{x}}=\fe{\cos}{x}\) \(\lzoo{x}{\fe{\sin}{u}}=\fe{\cos}{u}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\sin}{y}}=\fe{\cos}{y}\lz{y}{x}\)
\(\lzoo{x}{\fe{\cos}{x}}=-\fe{\sin}{x}\) \(\lzoo{x}{\fe{\cos}{u}}=-\fe{\sin}{u}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\cos}{y}}=-\fe{\sin}{y}\lz{y}{x}\)
\(\lzoo{x}{\fe{\tan}{x}}=\fe{\sec^2}{x}\) \(\lzoo{x}{\fe{\tan}{u}}=\fe{\sec^2}{u}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\tan}{y}}=\fe{\sec^2}{y}\lz{y}{x}\)
\(\lzoo{x}{\fe{\sec}{x}}=\fe{\sec}{x}\fe{\tan}{x}\) \(\lzoo{x}{\fe{\sec}{u}}=\fe{\sec}{u}\fe{\tan}{u}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\sec}{y}}=\fe{\sec}{y}\fe{\tan}{y}\lz{y}{x}\)
\(\lzoo{x}{\fe{\cot}{x}}=-\fe{\csc^2}{x}\) \(\lzoo{x}{\fe{\cot}{u}}=-\fe{\csc^2}{u}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\cot}{y}}=-\fe{\csc^2}{y}\lz{y}{x}\)
\(\lzoo{x}{\fe{\csc}{x}}=-\fe{\csc}{x}\fe{\cot}{x}\) \(\lzoo{x}{\fe{\csc}{u}}=-\fe{\csc}{u}\fe{\cot}{u}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\csc}{y}}=-\fe{\csc}{y}\fe{\cot}{y}\lz{y}{x}\)
\(\lzoo{x}{\fe{\tan^{-1}}{x}}=\frac{1}{1+x^2}\) \(\lzoo{x}{\fe{\tan^{-1}}{u}}=\frac{1}{1+u^2}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\tan^{-1}}{y}}=\frac{1}{1+y^2}\lz{y}{x}\)
\(\lzoo{x}{\fe{\sin^{-1}}{x}}=\frac{1}{\sqrt{1-x^2}}\) \(\lzoo{x}{\fe{\sin^{-1}}{u}}=\frac{1}{\sqrt{1-u^2}}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\sin^{-1}}{y}}=\frac{1}{\sqrt{1-y^2}}\lz{y}{x}\)
\(\lzoo{x}{\fe{\sec^{-1}}{x}}=\frac{1}{\abs{x}\sqrt{x^2-1}}\) \(\lzoo{x}{\fe{\sec^{-1}}{u}}=\frac{1}{\abs{u}\sqrt{u^2-1}}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\sec^{-1}}{y}}=\frac{1}{\abs{y}\sqrt{y^2-1}}\lz{y}{x}\)
\(\lzoo{x}{e^x}=e^x\) \(\lzoo{x}{e^u}=e^u\lzoo{x}{u}\) \(\lzoo{x}{e^y}=e^y\lz{y}{x}\)
\(\lzoo{x}{a^x}=\fe{\ln}{a}a^x\) \(\lzoo{x}{a^u}=\fe{\ln}{a}a^u\lzoo{x}{u}\) \(\lzoo{x}{a^y}=\fe{\ln}{a}a^y\lz{y}{x}\)
\(\lzoo{x}{\fe{\ln}{x}}=\frac{1}{x}\) \(\lzoo{x}{\fe{\ln}{u}}=\frac{1}{u}\lzoo{x}{u}\) \(\lzoo{x}{\fe{\ln}{y}}=\frac{1}{y}\lz{y}{x}\)
\(\lzoo{x}{\abs{x}}=\frac{\abs{x}}{x}\) \(\lzoo{x}{\abs{u}}=\frac{\abs{u}}{u}\lzoo{x}{u}\) \(\lzoo{x}{\abs{y}}=\frac{\abs{y}}{y}\lz{y}{x}\)
TableB.0.13