Derivative of Inverse Hyperbolic Functions

Inverse hyperbolic functions have well-defined derivatives that indicate their differentiability within their domains. The derivatives of these functions are essential in calculus and mathematical analysis. The key inverse hyperbolic functions and their derivatives are: Inverse Hyperbolic Sine: \( \sinh^{-1}(x) \) Inverse Hyperbolic Cosine: \( \cosh^{-1}(x) \) Inverse Hyperbolic Tangent: \( \tanh^{-1}(x) \)

The formulas for the derivatives of the inverse hyperbolic functions are: \( \frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}} \) \( \frac{d}{dx}(\cosh^{-1}(x)) = \frac{1}{\sqrt{x^2 - 1}} \) \( \frac{d}{dx}(\tanh^{-1}(x)) = \frac{1}{1 - x^2} \) These formulas are valid within their respective domains.

We can derive the derivatives of inverse hyperbolic functions using implicit differentiation and trigonometric identities. Here are some methods: By Implicit Differentiation Using Hyperbolic Identities We will demonstrate the differentiation of \( \sinh^{-1}(x) \) using implicit differentiation: By Implicit Differentiation Assume \( y = \sinh^{-1}(x) \), so \( x = \sinh(y) \). Differentiate both sides with respect to x: \( \frac{d}{dx}(x) = \cosh(y) \cdot \frac{dy}{dx} \) Since \( \cosh^2(y) - \sinh^2(y) = 1 \), \( \cosh(y) = \sqrt{x^2 + 1} \). \( 1 = \sqrt{x^2 + 1} \cdot \frac{dy}{dx} \) implies \( \frac{dy}{dx} = \frac{1}{\sqrt{x^2 + 1}} \). Hence, \( \frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}} \).

Higher-order derivatives provide insights into the behavior of inverse hyperbolic functions. The first derivative shows the rate of change, while the second derivative indicates concavity or convexity. Calculating higher-order derivatives involves repeated differentiation of the first derivative. For example, the second derivative of \( \sinh^{-1}(x) \) is obtained from the first derivative \( \frac{1}{\sqrt{x^2 + 1}} \).

For \( x = 0 \), the derivative of \( \sinh^{-1}(x) \) is \( \frac{1}{\sqrt{0^2 + 1}} = 1 \). For \( x = 1 \), the derivative of \( \tanh^{-1}(x) \) is undefined because it reaches an asymptote there.

Inverse Hyperbolic Functions: Functions that are inverses of hyperbolic functions, such as \( \sinh^{-1}(x) \), \( \cosh^{-1}(x) \), and \( \tanh^{-1}(x) \). Implicit Differentiation: A method to find derivatives by differentiating both sides of an equation with respect to a variable. Chain Rule: A rule for differentiating composite functions by multiplying the derivatives of the inner and outer functions. Hyperbolic Identities: Equations such as \( \cosh^2(y) - \sinh^2(y) = 1 \), used in differentiating hyperbolic functions. Quotient Rule: A formula to differentiate a quotient of two functions, expressed as \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \).