Problem 56 - Calculus Single Variable

Question

Let \(f(x)=x /\left(1+x^{2}\right)^{2} .\) Use the identity $$ \frac{x /\left(1+x^{2}\right)^{2}-c /\left(1+c^{2}\right)^{2}}{x-c}=\frac{1-c x^{3}-c^{2} x^{2}-2 x c-x c^{3}}{\left(1+x^{2}\right)^{2}\left(1+c^{2}\right)^{2}} $$ to compute \(f^{\prime}(c)\).

Step-by-Step Solution

Answer

The derivative is \( f'(c) = \frac{1 - 3c^4 - 2c^2}{(1+c^2)^4} \).

Step 1: Prepare the Derivative Calculation

To find the derivative \(f'(c)\), we need to recognize the given identity as a difference quotient. The derivative \(f'(c)\) is the limit of this difference quotient as \(x \to c\). Thus, we have: \[f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} = \lim_{x \to c} \frac{x /(1+x^2)^2 - c/(1+c^2)^2}{x-c}.\]

Step 2: Analyzing the Identity

The identity given is \[\frac{x /(1+x^2)^2 - c/(1+c^2)^2}{x-c} = \frac{1 - c x^3 - c^2 x^2 - 2xc - xc^3}{(1+x^2)^2 (1+c^2)^2}.\]This identity is already in the form of a difference quotient on the left-hand side, and the right-hand side is a rational function. We want to find \(f'(c)\), which implicitly suggests evaluating the limit of the right-hand side as \(x \to c\).

Step 3: Evaluate the Limit

As \(x \to c\), the expression on the right-hand side becomes:\[\lim_{x \to c} \frac{1 - c x^3 - c^2 x^2 - 2xc - xc^3}{(1+x^2)^2 (1+c^2)^2}.\]Substitute \(x = c\) into the numerator:\[1 - c c^3 - c^2 c^2 - 2cc - cc^3.\]This simplifies to:\[1 - c^4 - c^4 - 2c^2 - c^4 = 1 - 3c^4 - 2c^2.\]Hence, \(f'(c)\) is:\[f'(c) = \frac{1 - 3c^4 - 2c^2}{(1+c^2)^4}.\]

Step 4: Simplify the Expression

After simplifying the limit, the expression for \(f'(c)\) becomes:\[f'(c) = \frac{1 - 3c^4 - 2c^2}{(1+c^2)^4}.\] This represents the derivative of the function \(f(x)\) with respect to \(x\) at the point \(x = c\).

Key Concepts

Difference Quotient

The difference quotient is a crucial concept in calculus, serving as the backbone for finding derivatives of functions. Think of it as a formula that reveals the rate of change of a function. For a given function \( f(x) \), the difference quotient is expressed as \( \frac{f(x) - f(c)}{x - c}\), where \(c\) is a specific point and \(x\) approaches \(c\). This formula essentially calculates the slope of the secant line between two points on the function's curve.

In the provided exercise, the difference quotient is adeptly utilized with a given identity:

• Numerator: \( x/(1+x^2)^2 - c/(1+c^2)^2 \)
• Denominator: \( x-c \)

Transforming this fraction helps us approach the derivative calculation, setting the stage for evaluating the limit as \(x\) approaches \(c\). Understanding the difference quotient is pivotal for solving calculus problems as it leads to identifying the function's derivative.

Limit Evaluation

Evaluating limits is a fundamental calculus technique, employed to find the value that a function approaches as its input closes in on a particular point. In the context of derivative calculation, limit evaluation is indispensable as it solidifies the concept of instantaneous rate of change.

In this exercise, limit evaluation comes into play primarily at the point where \(x\) approaches \(c\). The step-by-step solution shows the transformation from a complex difference quotient to a more manageable rational function:

• We analyze the expression: \( \lim_{x \to c} \frac{1 - c x^3 - c^2 x^2 - 2xc - xc^3}{(1+x^2)^2 (1+c^2)^2} \)

Substituting \(x = c\) into this expression lets us find the function's behavior at \(c\). This substitution simplifies the numerator, resulting in a new, simpler expression to evaluate. By mastering limit evaluation, students can unravel how functions behave near specific points, a key skill in calculus.

Rational Function

Rational functions are ratios of two polynomials, and they are common subjects in calculus problem solving due to their complexity and nuanced behavior. In this exercise, the difference quotient leads to a rational function on the right-hand side of the provided identity:

\( \frac{1 - c x^3 - c^2 x^2 - 2xc - xc^3}{(1+x^2)^2 (1+c^2)^2} \)

Rational functions like this one require careful handling, especially when simplifying and evaluating limits. Their behavior can change drastically with small variations in \(x\). By focusing on the rational function, the expression is simplified by substituting \(x = c\) in the numerator, which allows us to get the derivative \(f'(c)\) and fully grasp the function's dynamics at \(c\).

Understanding rational functions are pivotal for calculus students, helping them tackle complex problems involving fractions and polynomial expressions.

Calculus Problem Solving

Solving calculus problems often involves a blend of different concepts working in harmony. In this exercise, the process of finding the derivative \(f'(c)\) draws upon several core calculus skills including the use of difference quotients, limit evaluation, and rational functions.

This process can be broken down into a few manageable steps:

• Recognizing the structure of the difference quotient.
• Using limits to transition from the difference quotient to the derivative.
• Manipulating rational functions to simplify expressions and solve for derivatives.

By understanding these interconnected steps and how they relate to calculus principles, students can better navigate the complexity of derivative calculations. Problem-solving in calculus is not just about finding an answer, but also about appreciating the journey through logic and mathematical reasoning. This approach not only aids in this exercise but also provides a foundation for tackling new and more complex calculus challenges.