Problem 28 - Calculus Single Variable

Question

Calculate the requested derivative. \(\frac{d^{2} f}{d x^{2}}\) where \(f(x)=\frac{x^{2}}{x-1}\)

Step-by-Step Solution

Answer

The second derivative is \(\frac{2x^3 - 6x^2 + 4x}{(x-1)^4}\).

Step 1: Understanding the Function

The given function is a rational function, \(f(x) = \frac{x^2}{x-1}\). We need to find the second derivative, \(\frac{d^2 f}{dx^2}\), which requires determining the first derivative first.

Step 2: First Derivative Using Quotient Rule

The quotient rule for derivatives states that \(\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}\), where \(u = x^2\) and \(v = x-1\). Calculate \(u' = 2x\) and \(v' = 1\). The first derivative \( f'(x) = \frac{(x-1)(2x) - x^2(1)}{(x-1)^2} \).

Step 3: Simplify the First Derivative

Simplify \(f'(x) = \frac{2x^2 - 2x - x^2}{(x-1)^2} = \frac{x^2 - 2x}{(x-1)^2}\).

Step 4: Second Derivative Using Quotient Rule Again

Apply the quotient rule again to \(f'(x) = \frac{x(x-2)}{(x-1)^2}\). Here, \(u = x(x-2)\) and \(v = (x-1)^2\). First, calculate the derivatives: \(u' = x - 2 + x = 2x - 2\) and \(v' = 2(x-1)\).

Step 5: Calculate the Second Derivative

Plug the derivatives into the quotient rule: \(\frac{d^2 f}{dx^2} = \frac{(x-1)^2(2x-2) - x(x-2) \cdot 2(x-1)}{(x-1)^4}\).

Step 6: Simplify the Second Derivative

Simplify the expression: \(\frac{d^2 f}{dx^2} = \frac{(2x^3 - 6x^2 + 4x)}{(x-1)^4}\). After canceling terms and factoring, simplify to obtain a clearer expression.

Key Concepts

Quotient Rule

The Quotient Rule is an essential tool in calculus for differentiating functions that are divided by each other. When you have a function presented as a quotient, such as \( \frac{u}{v} \), it requires specific handling to find the derivative. The rule helps us to efficiently compute the derivative of a division of two differentiable functions.
Here's the formula:

• The derivative of \( \frac{u}{v} \) is given by \( \frac{v \cdot u' - u \cdot v'}{v^2} \).

This means that you:

• First differentiate the numerator \( u \) to get \( u' \) and the denominator \( v \) to get \( v' \).
• Multiply \( u' \) by \( v \) and \( v' \) by \( u \).
• Subtract the second product from the first one, and finally divide by the square of the denominator, \( v^2 \).

Applying the Quotient Rule correctly ensures we account for all changes in both the numerator and denominator, giving us the accurate rate of change of the entire rational function.

Rational Function

A rational function, like \( f(x) = \frac{x^2}{x-1} \), is essentially a ratio of two polynomials. These types of functions typically take the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

Rational functions can exhibit interesting behaviors, such as asymptotes, which are values that the function approaches but never actually reaches. Because the function involves division, we must be cautious about where the denominator is zero, as these are the values where the function is undefined.

In our function \( f(x) = \frac{x^2}{x-1} \), the denominator \( x-1 \) becomes zero when \( x = 1 \). Hence, there's a vertical asymptote at \( x = 1 \). This behavior is crucial for understanding the overall curve of the function and how it shifts as \( x \) changes.

Differentiation Steps

The process of differentiation involves finding the derivative of a function to understand how it changes. It's a multi-step process when dealing with complex functions such as rational functions. Here’s how you approach differentiation in steps:

• Step 1: Simplify the Function - Before taking any derivatives, inspect the function and simplify it if possible. This helps in reducing errors and making calculations easier.


• Step 2: Apply the Differentiation Rule - For rational functions, you typically use the Quotient Rule. This involves identifying the numerator \( u(x) \) and the denominator \( v(x) \) separately. Compute their individual derivatives, then apply the quotient rule formula.
• Step 3: Simplify the Derivative - After applying the derivative rules, simplify the expression to make further calculations easier.
• Step 4: Calculate Higher-Order Derivatives - If needed, take the derivative of the derivative to get higher-order derivatives, like the second derivative, repeating the use of differentiation rules.

By following these structured steps, you ensure a logical and thorough approach to differentiation, reducing complexity and minimizing mistakes.