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Derivatives, Integrals, and Identities of Hyperbolic and Inverse Hyperbolic Functions

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  • Derivative of inverse sine function
    The derivative of \(\sin^{-1} u\) is \(\frac{du/dx}{\sqrt{1-u^2}}\), valid for \(|u|<1\).
  • Derivative of inverse cosine function
    The derivative of \(\cos^{-1} u\) is \(-\frac{du/dx}{\sqrt{1-u^2}}\), valid for \(|u|<1\).
  • Derivative of inverse tangent function
    The derivative of \(\tan^{-1} u\) is \(\frac{du/dx}{1+u^2}\).
  • Definition of hyperbolic sine and cosine
    sinh x = \(\frac{e^x - e^{-x}}{2}\), cosh x = \(\frac{e^x + e^{-x}}{2}\).
  • Hyperbolic tangent and cotangent definitions
    tanh x = \(\frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\), coth x = \(\frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}\).
  • Key hyperbolic identities
    \(\cosh^2 x - \sinh^2 x = 1\), \(\sinh 2x = 2 \sinh x \cosh x\), \(\cosh 2x = \cosh^2 x + \sinh^2 x\).
  • Derivative of sinh u
    \(\frac{d}{dx} (\sinh u) = \cosh u \frac{du}{dx}\).
  • Derivative of cosh u
    \(\frac{d}{dx} (\cosh u) = \sinh u \frac{du}{dx}\).
  • Derivative of tanh u
    \(\frac{d}{dx} (\tanh u) = \operatorname{sech}^2 u \frac{du}{dx}\).
  • Derivative of coth u
    \(\frac{d}{dx} (\coth u) = -\operatorname{csch}^2 u \frac{du}{dx}\).
  • Integral of sinh u
    \(\int \sinh u \, du = \cosh u + C\).
  • Integral of cosh u
    \(\int \cosh u \, du = \sinh u + C\).
  • Integral of sech² u
    \(\int \operatorname{sech}^2 u \, du = \tanh u + C\).
  • Inverse hyperbolic function relation for sech⁻¹ and csch⁻¹
    \(\operatorname{sech}^{-1} x = \cosh^{-1} \frac{1}{x}\), \(\operatorname{csch}^{-1} x = \sinh^{-1} \frac{1}{x}\).
  • Derivative of inverse sinh function
    \(\frac{d}{dx} (\sinh^{-1} u) = \frac{1}{\sqrt{1+u^2}} \frac{du}{dx}\).
  • Derivative of inverse cosh function
    \(\frac{d}{dx} (\cosh^{-1} u) = \frac{1}{\sqrt{u^2 - 1}} \frac{du}{dx}\), valid for \(u > 1\).
  • Derivative of inverse tanh function
    \(\frac{d}{dx} (\tanh^{-1} u) = \frac{1}{1 - u^2} \frac{du}{dx}\), valid for \(|u| < 1\).
  • Integral involving inverse hyperbolic sine
    \(\int \frac{du}{\sqrt{a^2 + u^2}} = \sinh^{-1} \left( \frac{u}{a} \right) + C, \quad a > 0\).
  • Logarithmic form of inverse sinh
    \(\sinh^{-1} x = \ln \left( x + \sqrt{x^2 + 1} \right)\).
  • Logarithmic form of inverse tanh
    \(\tanh^{-1} x = \frac{1}{2} \ln \frac{1+x}{1-x}\), valid for \(|x| < 1\).