Problem 54 - Essential Calculus

Question

Find the derivative of the function. \(y=\tanh ^{-1} \frac{x}{2}\)

Step-by-Step Solution

Answer

The derivative of the function \(y=\tanh^{-1} \frac{x}{2}\) is \(y' = \frac{2}{4 - x^2}\).

Step 1: Identify the function on which to apply the derivative

The given function is \(y=\tanh^{-1} \frac{x}{2}\). The derivative will be applied on it.

Step 2: Apply the derivative

The derivative of \(y = \tanh^{-1} u\) is \(\frac{1}{1-u^2}\). Apply this to the given function such that \(u = \frac{x}{2}\). This gives, \(y' = \frac{1}{1-\left(\frac{x}{2}\right)^2}\). Do not forget to apply the chain rule. The derivative of \(\frac{x}{2}\) is \(\frac{1}{2}\). So, the derivative of \(y\) becomes, \(y' = \frac{1}{2(1-\left(\frac{x}{2}\right)^2)}\)

Step 3: Simplify the expression

We have, \(y' = \frac{1}{2(1-\left(\frac{x}{2}\right)^2)} = \frac{1}{2\left(1-\frac{x^2}{4}\right)} = \frac{1}{2 - \frac{x^2}{2}} = \frac{2}{4 - x^2}\).

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are mathematical functions closely related to the standard hyperbolic functions, like sinh, cosh, and tanh. They are useful in solving equations involving hyperbolic functions, providing outputs that correspond to angles or hyperbolic angles.

• The inverse hyperbolic tangent function, written as \( \tanh^{-1} x \), is one of these functions.
• It's the inverse of the hyperbolic tangent function, \( \tanh x \), which is the ratio of sinh and cosh.
• In calculus, these functions are integral because of their unique derivatives and properties.

Inverse hyperbolic functions like \( \tanh^{-1} x \) help us work through complex differentiation problems involving hyperbolic terms, unveiling their usefulness in various mathematical and practical applications.

Chain Rule

In calculus, the chain rule is a fundamental tool for differentiating composite functions. It enables us to find the derivative of a function that comprises an inner and an outer function.

• Think of the chain rule as a method to "unpack" the layers of a composition in order to differentiate it.
• Mathematically, it's expressed as \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).
• This implies you first differentiate the outer function, leaving the inner function untouched, then multiply it by the derivative of the inner function.
• In the problem \( y = \tanh^{-1} \frac{x}{2} \), the inner function is \( \frac{x}{2} \) and the outside function is \( \tanh^{-1} u \).

By applying the chain rule correctly, we can handle derivatives of nested functions efficiently, breaking complex tasks into simpler, manageable parts.

Differentiation

Differentiation is the central process in calculus that deals with finding the rate at which one quantity changes with respect to another. It's essentially the process of finding the derivative.

• The derivative of a function at a point provides the slope of the tangent to the curve at that point.
• It's used across many fields such as physics for motion, biology for growth rates, and economics for cost functions.
• In our example, finding \( y' \) involves differentiating \( \tanh^{-1} \frac{x}{2} \) using the formula for the derivative of the inverse hyperbolic tangent.
• Simplification often follows differentiation, ensuring the expression is as clear and compact as possible.

Understanding differentiation's core principles allows you to tackle a variety of mathematical problems, bridging the gap between static values and dynamic changes.