Problem 405 - Calculus Volume 1

Question

For the following exercises, find the derivatives for the functions. $$\tanh ^{-1}(4 x)$$

Step-by-Step Solution

Answer

The derivative of \( \tanh^{-1}(4x) \) is \( \frac{4}{1-16x^2} \).

Step 1: Understand the Function

The function we need to find the derivative for is \( f(x) = \tanh^{-1}(4x) \), which represents the inverse hyperbolic tangent function. Our task is to find \( f'(x) \).

Step 2: Recall the Derivative of Inverse Hyperbolic Tangent

The derivative of the inverse hyperbolic tangent function, \( \tanh^{-1}(u) \), with respect to \( u \), is given by \( \frac{d}{du}[\tanh^{-1}(u)] = \frac{1}{1-u^2} \) provided \( |u| < 1 \).

Step 3: Apply the Chain Rule

Since we have \( \tanh^{-1}(4x) \), let \( u = 4x \). We apply the chain rule to find \( \frac{d}{dx}[\tanh^{-1}(4x)] = \frac{d}{du}[\tanh^{-1}(u)] \cdot \frac{du}{dx} \).

Step 4: Differentiate the Inner Function

First, differentiate \( u = 4x \) with respect to \( x \). Thus, \( \frac{du}{dx} = 4 \).

Step 5: Combine Results Using the Chain Rule

Substitute \( \frac{du}{dx} = 4 \) and \( \frac{d}{du}[\tanh^{-1}(u)] = \frac{1}{1-u^2} \) into the chain rule formula: \[ f'(x) = \frac{1}{1-(4x)^2} \cdot 4 = \frac{4}{1-16x^2} \]

Step 6: Simplify the Result

The derivative cannot be simplified further, so the final answer is: \[ f'(x) = \frac{4}{1-16x^2} \]

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are analogs of inverse trigonometric functions, but they are linked to hyperbolic functions. One of the key functions in this family is the inverse hyperbolic tangent, denoted as \( anh^{-1}(x)\). This function is useful in various mathematical contexts, especially when dealing with hyperbolic equations. The inverse hyperbolic tangent function can be defined in terms of a logarithm:

• \(\tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)\)

This formula makes it clear why the function is only defined for \(|x| < 1\), similar to the restrictions you would find in inverse trigonometric functions.
Studying inverse hyperbolic functions is important because they arise in many areas of calculus and even in complex analysis. In practical applications, they often help model phenomena like gravitational calculations or are used in solving differential equations.

Chain Rule

The chain rule is a fundamental concept in calculus used for finding the derivatives of composite functions. If you have a function \(f(x) = g(h(x))\), then the chain rule tells you that the derivative \(f'(x)\) is calculated as:

• \(f'(x) = g'(h(x)) \cdot h'(x)\)

This principle helps us break down complex functions into simpler parts. In practice, we:

• Differentiate the outer function \(g\)
• Evaluate it at the inner function \(h(x)\)
• Multiply the result by the derivative of the inner function \(h'(x)\)

In the given exercise, the internal function is \(4x\), and the external function is \(\tanh^{-1}(u)\) where \(u = 4x\). By applying the chain rule, we simplified the process of finding the derivative of this composite function.Using the chain rule effectively requires practice, but it is a powerful tool that makes calculating derivatives manageable and efficient.

Derivatives

Derivatives are a key concept in calculus, representing the rate at which a function changes at any given point. They are essential for understanding behavior and patterns in both theoretical and applied mathematics. In simple terms, the derivative provides information about the slope or steepness of a curve described by a function.For inverse hyperbolic functions like \(\tanh^{-1}(x)\), the derivative has a specific formula:

• \(\frac{d}{dx}[\tanh^{-1}(u)] = \frac{1}{1-u^2}\)

This formula only holds when \(|u| < 1\) to ensure the function remains defined.
Applying this derivative, alongside tools like the chain rule, allows us to solve complex differentiation problems, as illustrated in the exercise. The final derivative \(\frac{4}{1-16x^2}\) indicates how sharply the inverse hyperbolic tangent function changes relative to \(x\). Recognizing derivatives helps us understand dynamic systems in physics, engineering, and beyond.