Tutorial 8: Hyperbolic Functions

Hyperbolic Functions: Key Concepts

Hyperbolic Functions

• Defined using exponential functions:
  • $\sinh x = \frac{e^x - e^{-x}}{2}$
  • $\cosh x = \frac{e^x + e^{-x}}{2}$
  • $\tanh x = \frac{\sinh x}{\cosh x}$
• Identities mirror trigonometric identities:
  • $\cosh^2 x - \sinh^2 x = 1$
  • $\sinh(2x) = 2 \sinh x \cosh x$

Derivatives

• Basic derivatives:
  • $\frac{d}{dx} \sinh x = \cosh x$
  • $\frac{d}{dx} \cosh x = \sinh x$
  • $\frac{d}{dx} \tanh x = \text{sech}^2 x$
• Chain rule examples:
  • $y = \sinh(4x - 8) \implies \frac{dy}{dx} = 4 \cosh(4x - 8)$
  • $y = \cosh(x^4) \implies \frac{dy}{dx} = 4x^3 \sinh(x^4)$
  • $y = \sinh^3(2x) \implies \frac{dy}{dx} = 6 \sinh^2(2x) \cosh(2x)$

Integrals

• Standard integrals:
  • $\int \sinh x , dx = \cosh x + C$
  • $\int \cosh x , dx = \sinh x + C$
• Substitution method:
  • $\int \sinh^6 x \cosh x , dx = \frac{\sinh^7 x}{7} + C$ (let $u = \sinh x$)
  • $\int \cosh(2x - 3) , dx = \frac{1}{2} \sinh(2x - 3) + C$

Inverse Hyperbolic Functions

• Defined as logarithms:
  • $\sinh^{-1} x = \ln\left(x + \sqrt{1 + x^2}\right)$
  • $\cosh^{-1} x = \ln\left(x + \sqrt{x^2 - 1}\right)$, $x \geq 1$
  • $\tanh^{-1} x = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)$, $|x| < 1$
• Properties:
  • $\cosh^{-1} x$ is defined only for $x \geq 1$
  • $\tanh^{-1} 0 = 0$
• Derivative forms:
  • $\frac{d}{dx} \sinh^{-1} x = \frac{1}{\sqrt{1 + x^2}}$
  • $\frac{d}{dx} \cosh^{-1} x = \frac{1}{\sqrt{x^2 - 1}}$