Problem 44 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=x^{2} \sqrt{x^{2}-1} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{3x^3 - 2x}{\sqrt{x^2 - 1}} \).

Step 1: Rewrite the Function

First, we need to rewrite the function to a form more easily differentiated. Given the function is \( f(x) = x^2 \sqrt{x^2 - 1} \), let's express \( \sqrt{x^2 - 1} \) as a power: \( (x^2 - 1)^{1/2} \). Thus, the function becomes: \[ f(x) = x^2 (x^2 - 1)^{1/2} \].

Step 2: Apply the Product Rule

Since the function \( f(x) = x^2 (x^2 - 1)^{1/2} \) is a product of two functions, we use the product rule. The product rule is given by \( (uv)' = u'v + uv' \), where \( u = x^2 \) and \( v = (x^2 - 1)^{1/2} \).

Step 3: Differentiate Each Function

Differentiate each part: - \( u = x^2 \) gives \( u' = 2x \).- \( v = (x^2 - 1)^{1/2} \) requires the chain rule. Let \( g(x) = x^2 - 1 \), then \( v = g(x)^{1/2} \), so \( v' = \frac{1}{2}g(x)^{-1/2} \cdot g'(x) \). Now calculate \( g'(x) = 2x \), so \( v' = \frac{1}{2}(x^2 - 1)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 - 1}} \).

Step 4: Substitute into the Product Rule

Substitute the derivatives found in the previous step into the product rule: \[ f'(x) = (2x) (x^2 - 1)^{1/2} + x^2 \left(\frac{x}{\sqrt{x^2 - 1}}\right) \].Simplify the expression:\[ f'(x) = 2x(x^2 - 1)^{1/2} + \frac{x^3}{\sqrt{x^2 - 1}} \].

Step 5: Simplify the Expression

To consolidate the terms, get a common denominator for the second term:\[ f'(x) = 2x \sqrt{x^2 - 1} + \frac{x^3}{\sqrt{x^2 - 1}} \].Factor out \( \sqrt{x^2 - 1} \): \[ f'(x) = \frac{2x(x^2 - 1) + x^3}{\sqrt{x^2 - 1}} = \frac{2x^3 - 2x + x^3}{\sqrt{x^2 - 1}} \].Combine like terms:\[ f'(x) = \frac{3x^3 - 2x}{\sqrt{x^2 - 1}} \].

Key Concepts

Generalized Power Rule

The Generalized Power Rule is an extension of the traditional power rule in calculus, which allows us to differentiate functions of the form \( x^n \), where \( n \) is any real number, not just integers. This becomes particularly useful when dealing with roots or more complex exponents.

Given a function in the form of \( (h(x))^n \), the Generalized Power Rule states that:

• Differentiating \( (h(x))^n \) involves multiplying \( n \) by \( (h(x))^{n-1} \).
• You then multiply this by the derivative of the inner function \( h'(x) \).

In simpler terms, the formula for the Generalized Power Rule is:
\[\frac{d}{dx}(h(x))^n = n(h(x))^{n-1} \cdot h'(x)\]
In the given exercise, the square root \( \sqrt{x^2 - 1} \) is rewritten as \( (x^2-1)^{1/2} \) to make it easier to apply this rule.

Product Rule

The Product Rule is a differentiation technique used when you need to find the derivative of a function that is the product of two other functions. If you have a function \( f(x) = u(x)v(x) \), the derivative \( f'(x) \) is given by:

• \( u'v + uv' \)

This means you take the derivative of the first function and multiply by the second function as it is, then add the product of the first function and the derivative of the second function.

In our exercise, we identified \( u = x^2 \) and \( v = (x^2 - 1)^{1/2} \). The product rule was applied in step 3:

• First, calculate \( u' = 2x \)
• Then, use the Chain Rule (explained below) to find \( v' = \frac{x}{\sqrt{x^2 - 1}} \)

Finally, substitute these into the product rule formula to compute \( f'(x) \).

Chain Rule

The Chain Rule is essential for differentiating composite functions, functions within other functions. It explains how to take the derivative of a function that is composed of two or more functions.

The Chain Rule is structured like this: if you have a function \( f(g(x)) \), the derivative \( f'(x) \) is \( f'(g(x)) \cdot g'(x) \). This means you differentiate the outer function, leave the inner function as it is, and then multiply by the derivative of the inner function.

In our example, the portion \( (x^2 - 1)^{1/2} \) uses the Chain Rule heavily. Here's the process:

• Set \( g(x) = x^2 - 1 \), thus \( v = g(x)^{1/2} \).
• First, find \( g'(x) = 2x \).
• Next, differentiate the outer expression: derivative of \( u^n \) is \( \frac{1}{2}g(x)^{-1/2} \).
• Multiply by \( g'(x) \) to get \( v' = \frac{x}{\sqrt{x^2 - 1}} \).

Derivative simplification

After finding the derivative using differentiation rules, simplification is vital for expressing the derivative in the cleanest form. It involves combining like terms, factoring, and adjusting expressions for easier understanding and further use.

In our solution, once the derivatives are calculated using the Product and Chain Rules, the resulting expression is:\[f'(x) = 2x(x^2 - 1)^{1/2} + \frac{x^3}{\sqrt{x^2 - 1}}\]
To streamline this expression, here are the steps applied:

• Factor the common expression or term, in our case, \( \sqrt{x^2 - 1} \).
• Combine the terms in the numerator by getting a common denominator.
• Write the simplified result: \( f'(x) = \frac{3x^3 - 2x}{\sqrt{x^2 - 1}} \).

These steps help in making the expression neater and ready for further mathematical applications.