Problem 55 - Calculus Single Variable

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Identity

We are given an identity: \[\frac{x /(1+x^{2})-c /(1+c^{2})}{x-c}=\frac{1-x c}{(1+x^{2})(1+c^{2})}\] This is a specific setup to find the derivative of the function \(f(x)\) at \(x=c\). The identity simplifies the expression for the difference quotient, which is useful for evaluating the derivative.

Step 2: Recall the Definition of the Derivative

The derivative of \(f(x)\) at a point \(c\) is defined as: \[f'(c) = \lim_{x \to c}\frac{f(x) - f(c)}{x - c}\] In this context, the equation from the problem provides the simplified form of the difference quotient for \(f(x) = x/(1 + x^2)\). Our job is to find \(\lim_{x \to c}\) of this expression.

Step 3: Substitute and Simplify

Plug the given identity into the derivative's limit formula:\[f'(c) = \lim_{x \to c}\frac{x/(1+x^2)-c/(1+c^2)}{x-c}\]From the problem, we know:\[\frac{x/(1+x^2)-c/(1+c^2)}{x-c} = \frac{1-xc}{(1+x^2)(1+c^2)}\] Thus:\[f'(c) = \lim_{x \to c}\frac{1-xc}{(1+x^2)(1+c^2)}\]

Step 4: Evaluate the Limit

Now evaluate the limit:\[\lim_{x \to c}\frac{1-xc}{(1+x^2)(1+c^2)}\]Substitute \(x = c\) directly since the expression is no longer in an indeterminate form:\[= \frac{1 - c^2}{(1 + c^2)(1 + c^2)} = \frac{1 - c^2}{(1 + c^2)^2}\] Thus, the derivative of the function \(f(x)\) at \(x = c\) is \(f'(c) = \frac{1 - c^2}{(1 + c^2)^2}\).

Key Concepts

Difference Quotient

The concept of the difference quotient is central to understanding derivatives. It allows us to compute the derivative by approximating the slope of the tangent line to a curve at a point. The difference quotient formula is given by: \[\frac{f(x) - f(c)}{x - c}\] This represents the average rate of change of the function \(f(x)\) between \(x\) and \(c\).

• In the context of the problem, the difference quotient we use is specific to the function \(f(x) = \frac{x}{1+x^2}\).
• This quotient becomes simpler with provided identities, making the derivative calculation more straightforward.

Understanding this concept is key as it serves as a building block for finding the limit to define the derivative.

Definition of the Derivative

The derivative of a function at a point measures how the function changes as its input changes. For a function \(f(x)\), the derivative at a point \(c\) is expressed as:\[f'(c) = \lim_{x \to c}\frac{f(x) - f(c)}{x - c}\]

• This expression means that the derivative is the limit of the difference quotient as \(x\) approaches \(c\).
• It is about finding the instantaneous rate of change or the slope of the tangent at that point.

In our problem, we substitute the identity we have for the difference quotient, focusing on simplifying and evaluating the limit to find the derivative.

Limit

Limits are fundamental in calculus. They help us understand the behavior of functions as they approach a particular point. The limit process is involved in defining derivatives and handling indeterminate forms. For a function, the limit as \(x\) approaches a specific value gives insight into the function's behavior around that point.

• In our problem, the limit \(\lim_{x \to c}\frac{1-xc}{(1+x^2)(1+c^2)}\) is evaluated when \(x\) approaches \(c\).
• Once eliminated from indeterminacy, this allows us to plug in \(x = c\) directly to find the derivative.

Understanding limits is essential in calculus as they outline the foundation for derivative calculations.

Function

Functions describe relationships where each input is associated with exactly one output. In calculus, functions allow us to model a wide range of phenomena. The function \(f(x) = \frac{x}{1+x^2}\) captures a specific mathematical relationship.

• This mathematical expression forms the foundation for building the derivative and limits that we examine in the problem.
• Understanding the structure of the function allows for substitutions and comparisons needed for derivative evaluation.

Recognizing how functions behave helps us in manipulating and applying rules like the difference quotient to find solutions effectively.