Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\operatorname{coth}^{-1}\left(x^{5}\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{5x^4}{1 - x^{10}} \).

Step 1: Recognize the Inverse Function Derivative Rule

We know the derivative of the inverse hyperbolic cotangent function, \( \operatorname{coth}^{-1}(u) \), is given by \( \frac{1}{1-u^2} \) for \( |u|>1 \). We will use this rule to differentiate the given function.

Step 2: Apply the Chain Rule

Since our function is \( \operatorname{coth}^{-1}(x^5) \), let \( u = x^5 \). Then, by the chain rule, the derivative of \( \operatorname{coth}^{-1}(u) \) is \( \frac{d}{dx} \left[ \operatorname{coth}^{-1}(u) \right] = \frac{1}{1-u^2} \times \frac{du}{dx} \).

Step 3: Differentiate the Inner Function

Differentiate \( u = x^5 \) with respect to \( x \). This gives \( \frac{du}{dx} = 5x^4 \).

Step 4: Substitute and Simplify

Now, plug \( u = x^5 \) and \( \frac{du}{dx} = 5x^4 \) into the derivative expression: \( D_{x} y = \frac{1}{1 - (x^5)^2} \times 5x^4 = \frac{5x^4}{1 - x^{10}} \).

Key Concepts

Inverse Function Differentiation

When we talk about finding the derivative of an inverse function, what we're doing is employing a special rule known as the "Inverse Function Differentiation Rule." This rule helps us find the rate at which an inverse function changes. In simple terms, the inverse function of a well-known function basically flips the roles of inputs and outputs.

For hyperbolic functions, like the inverse hyperbolic cotangent, the derivative might seem mysterious at first. But we can use the formula: if \( y = \operatorname{coth}^{-1}(u) \), then its derivative is \( \frac{1}{1 - u^2} \) when \(|u| > 1\). This formula emerges from calculus concepts about how flipping functions interact with their derivatives. Let’s remember that the restriction \(|u| > 1\) is crucial because it ensures the function is defined where we’re deriving it.

Chain Rule

The Chain Rule is a fundamental technique in calculus. It’s a way to tackle the derivative of a composite function. A composite function is just a fancy term for when one function is inside another, like a Russian nesting doll.

Let’s think about our exercise: we have \( y = \operatorname{coth}^{-1}(x^5) \). Here, \( x^5 \) is nested inside the \( \operatorname{coth}^{-1} \) function. To differentiate this, we can’t just look at one bit of the function at a time—we need to appreciate how they work together. That's where the Chain Rule shines!

• First, identify the outer function: \( \operatorname{coth}^{-1}(x) \)
• Then, identify the inner function: \( x^5 \)
• The Chain Rule says: multiply the derivative of the outer function, evaluated at the inner function, by the derivative of the inner function itself.

This gives us: \( \frac{1}{1-(x^5)^2} \times 5x^4 \), capturing how changes in \( x \) ripple through both the inner and outer functions.

Hyperbolic Functions

Hyperbolic functions are cousins of the trigonometric functions but are based on hyperbolas rather than circles. Some common ones include hyperbolic sine (\( \sinh \)), hyperbolic cosine (\( \cosh \)), and hyperbolic cotangent (\( \coth \)). These functions show up in calculations involving areas like engineering and physics, where they model phenomena from catenaries to electromagnetic fields.

Their inverses, like \( \operatorname{coth}^{-1} \), help us "unwrap" or "invert" hyperbolic transformations—kind of like logging off a function or backing out of an equation. The inverse hyperbolic cotangent \( \operatorname{coth}^{-1}(x) \) is specifically defined for values of \( x \) where \(|x| > 1\). This ensures we're working within the arena where our function makes sense and behaves nicely.

• Hyperbolic functions often have intuitive geometric interpretations, akin to "stretched out" sine and cosine waves.
• They connect deeply with exponential functions, due to their definitions involving \( e^x \) and \( e^{-x} \).

So, when you see something like \( \operatorname{coth}^{-1}(x^5) \), what's happening is a play between several mathematical themes where exponential, hyperbolic, and inverse concepts mix.