Derivative of Inverse Hyperbolic Functions
2026-02-21 20:36 Diff
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Last updated on August 5, 2025

The derivative of inverse hyperbolic functions is crucial in understanding how these functions change in response to variations in x. Derivatives are useful in various fields, from engineering to economics, where they help in optimizing solutions. We will now delve into the derivatives of inverse hyperbolic functions.

What is the 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: \( \tanh^{-1}(x) \)

Derivative of Inverse Hyperbolic Functions Formula

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.

Proofs of the Derivative of Inverse Hyperbolic Functions

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, \( \frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}} \).

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Higher-Order Derivatives of Inverse Hyperbolic Functions

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 \( \frac{1}{\sqrt{x^2 + 1}} \).

Special Cases:

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.

Common Mistakes and How to Avoid Them in Derivatives of Inverse Hyperbolic Functions

Students often make errors when differentiating inverse hyperbolic functions. Here are some common mistakes and solutions:

Problem 1

Calculate the derivative of \( (\sinh^{-1}(x) \cdot \cosh^{-1}(x)) \).

Okay, lets begin

Let \( f(x) = \sinh^{-1}(x) \cdot \cosh^{-1}(x) \). Using the product rule, \( f'(x) = u'v + uv' \). Here, \( u = \sinh^{-1}(x) \) and \( v = \cosh^{-1}(x) \). Differentiate each: \( u' = \frac{1}{\sqrt{x^2 + 1}} \), \( v' = \frac{1}{\sqrt{x^2 - 1}} \). Substituting, \( f'(x) = \left(\frac{1}{\sqrt{x^2 + 1}}\right) \cdot \cosh^{-1}(x) + \sinh^{-1}(x) \cdot \left(\frac{1}{\sqrt{x^2 - 1}}\right) \). Thus, \( f'(x) = \frac{\cosh^{-1}(x)}{\sqrt{x^2 + 1}} + \frac{\sinh^{-1}(x)}{\sqrt{x^2 - 1}} \).

Explanation

We apply the product rule to differentiate the given function. By dividing it into two parts, we differentiate each part separately and combine them using the product rule.

Well explained 👍

Problem 2

A cable is suspended between two poles, forming a catenary described by \( y = \cosh^{-1}(x) \). If x = 2 meters, find the slope of the cable.

Okay, lets begin

Given \( y = \cosh^{-1}(x) \), the slope is the derivative \( \frac{dy}{dx} = \frac{1}{\sqrt{x^2 - 1}} \). Substitute \( x = 2 \): \( \frac{dy}{dx} = \frac{1}{\sqrt{2^2 - 1}} = \frac{1}{\sqrt{3}} \). The slope of the cable at x = 2 meters is \( \frac{1}{\sqrt{3}} \).

Explanation

To find the slope of the cable, we differentiate the given catenary function and substitute \( x = 2 \) into the derivative to find the rate of change at that point.

Well explained 👍

Problem 3

Derive the second derivative of \( y = \tanh^{-1}(x) \).

Okay, lets begin

First, find the first derivative: \( \frac{dy}{dx} = \frac{1}{1 - x^2} \). Now, differentiate again for the second derivative: \( \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{1}{1 - x^2}\right) \). Using the chain rule, \( \frac{d^2y}{dx^2} = \frac{0 \cdot (1 - x^2) + 2x}{(1 - x^2)^2} = \frac{2x}{(1 - x^2)^2} \). Thus, the second derivative is \( \frac{2x}{(1 - x^2)^2} \).

Explanation

We differentiate the first derivative using the chain rule to find the second derivative, carefully handling the expression involving \( x \).

Well explained 👍

Problem 4

Prove: \( \frac{d}{dx}(\cosh^{-1}(x^2)) = \frac{2x}{\sqrt{x^4 - 1}} \).

Okay, lets begin

Start by applying the chain rule: Let \( y = \cosh^{-1}(x^2) \). Then, \( \frac{dy}{dx} = \frac{1}{\sqrt{x^4 - 1}} \cdot \frac{d}{dx}(x^2) \). \( \frac{d}{dx}(x^2) = 2x \). Substitute back: \( \frac{dy}{dx} = \frac{2x}{\sqrt{x^4 - 1}} \). Thus, proved.

Explanation

We apply the chain rule to differentiate \( \cosh^{-1}(x^2) \), multiplying the derivative of the outer function by the derivative of the inner function.

Well explained 👍

Problem 5

Solve: \( \frac{d}{dx}(\sinh^{-1}(x)/x) \).

Okay, lets begin

Use the quotient rule: \( \frac{d}{dx}\left(\frac{\sinh^{-1}(x)}{x}\right) = \frac{x \cdot \frac{1}{\sqrt{x^2 + 1}} - \sinh^{-1}(x) \cdot 1}{x^2} \). Simplify: \( = \frac{x}{x^2 \sqrt{x^2 + 1}} - \frac{\sinh^{-1}(x)}{x^2} \). Therefore, \( \frac{d}{dx}\left(\frac{\sinh^{-1}(x)}{x}\right) = \frac{x}{x^2 \sqrt{x^2 + 1}} - \frac{\sinh^{-1}(x)}{x^2} \).

Explanation

The quotient rule is used to differentiate the given function. By simplifying the expression, we arrive at the final result.

Well explained 👍

FAQs on the Derivative of Inverse Hyperbolic Functions

1.Find the derivative of \( \sinh^{-1}(x) \).

The derivative of \( \sinh^{-1}(x) \) is \( \frac{1}{\sqrt{x^2 + 1}} \).

2.Can the derivative of inverse hyperbolic functions be used in real life?

Yes, derivatives of inverse hyperbolic functions are used in fields like physics and engineering to model phenomena such as cables, chains, and other structures.

3.Is it possible to take the derivative of \( \cosh^{-1}(x) \) when \( x < 1 \)?

No, \( \cosh^{-1}(x) \) is undefined for \( x < 1 \), so the derivative cannot be taken in that range.

4.What rule is used to differentiate \( \sinh^{-1}(x^2) \)?

The chain rule is used to differentiate \( \sinh^{-1}(x^2) \), resulting in \( \frac{2x}{\sqrt{x^4 + 1}} \).

5.Are the derivatives of \( \sinh^{-1}(x) \) and \( \sinh(x) \) the same?

No, they are different. The derivative of \( \sinh^{-1}(x) \) is \( \frac{1}{\sqrt{x^2 + 1}} \), whereas the derivative of \( \sinh(x) \) is \( \cosh(x) \).

Important Glossaries for the Derivative of Inverse Hyperbolic Functions

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 \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \).

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Jaskaran Singh Saluja

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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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