Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function.\(f(x)=\frac{\sqrt{x^{2}+1}}{\sqrt{x^{2}-1}}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} - 1}}\) is \(f'(x) = \frac{-2x}{x^{2}-1}\).

Step 1: Recall the necessary rules

Before proceeding with the solution, recall the quotient rule and chain rule. Quotient rule for differentiation: If \(f(x) = \frac{g(x)}{h(x)}\), then \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\). Chain rule for differentiation: If \(y = f(g(x))\), then \(y' = f'(g(x)) \cdot g'(x)\).

Step 2: Apply the quotient rule

Apply the quotient rule for the given function \(f(x) = \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} - 1}}\), where \(g(x) = \sqrt{x^{2} + 1}\) and \(h(x) = \sqrt{x^{2} - 1}\). Therefore, \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\). Now we need to find \(g'(x)\) and \(h'(x)\).

Step 3: Compute the derivatives of g(x) and h(x)

Apply the chain rule to find \(g'(x)\) and \(h'(x)\). \(g(x) = \sqrt{x^{2} + 1} = (x^{2} + 1)^{1/2}\). So, \(g'(x) = \frac{1}{2}(x^{2} + 1)^{-1/2} \cdot (2x) = \frac{x}{\sqrt{x^{2} + 1}}\). \(h(x) = \sqrt{x^{2} - 1} = (x^{2} - 1)^{1/2}\). So, \(h'(x) = \frac{1}{2}(x^{2} - 1)^{-1/2} \cdot (2x) = \frac{x}{\sqrt{x^{2} - 1}}\).

Step 4: Plug the derivatives and functions into the quotient rule

Substitute the computed derivatives and functions into the quotient rule formula. \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\) = \(\frac{\frac{x}{\sqrt{x^{2} + 1}} \cdot \sqrt{x^{2} - 1} - \frac{x}{\sqrt{x^{2} - 1}} \cdot \sqrt{x^{2} + 1}}{(\sqrt{x^{2} - 1})^2}\).

Step 5: Simplify the expression

Simplify the expression to get the final answer. \(f'(x) = \frac{x(x^{2}-1)-x(x^{2}+1)}{(x^{2}-1)} = \frac{x^{3}-x-x^{3}-x}{(x^{2}-1)} = \frac{-2x}{x^{2}-1}\). So, the derivative of the function \(f(x) = \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} - 1}}\) is \(f'(x) = \frac{-2x}{x^{2}-1}\).

Key Concepts

Quotient Rule

The Quotient Rule is essential whenever you are differentiating a function that is the ratio of two other functions. In simpler terms, if you have a function expressed as one function divided by another, like \(f(x) = \frac{g(x)}{h(x)}\), you must use the Quotient Rule to find its derivative. This rule states:

• Differentiate the numerator function \(g(x)\) to get \(g'(x)\).
• Differentiate the denominator function \(h(x)\) to get \(h'(x)\).
• Apply the formula: \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\).

This rule helps maintain the structure of the derivative, even when dealing with complex divisions. Always remember: quotient means division, so think of it as the process of 'chopping up' a fraction to find the rate of change.

Chain Rule

The Chain Rule is a fundamental technique for finding the derivative of compositions of functions. When one function is nestled inside another, like \(y = f(g(x))\), the Chain Rule comes into play. Here's how it works:

• Take the derivative of the outer function, treating the inner function as a single variable.
• Then multiply this result by the derivative of the inner function.

In mathematical terms: if you have \(y = f(g(x))\), then the derivative \(y' = f'(g(x)) \cdot g'(x)\). This ensures that each function within your composite function is properly considered, adjusting for the effect of its rate of change. In our exercise, this is particularly used for finding the derivatives of \(\sqrt{x^2 + 1}\) and \(\sqrt{x^2 - 1}\), which are compositions of power and polynomial functions.

Derivative of Functions

Differentiating functions is all about finding the rate at which the function's value is changing at any given point. It provides insight into the behavior of functions, like growth or decay. For any given function \(f(x)\), its derivative \(f'(x)\) tells you how \(f(x)\) changes with a small change in \(x\).

• This concept forms the backbone of calculus and is crucial for calculating slopes of tangents to curves.
• Finding derivatives involves applying specific rules, such as the Power Rule, Product Rule, Quotient Rule, and Chain Rule, among others.

In the exercise you are tackling, finding the derivative of the function \(f(x) = \frac{\sqrt{x^2 + 1}}{\sqrt{x^2 - 1}}\) involves neatly combining these rules to achieve the correct derivative expression \(f'(x) = \frac{-2x}{x^2-1}\). Understanding each differentiation rule's purpose and application is key to mastering calculus.