Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{x}{x+2} $$

Step-by-Step Solution

Answer

\(f'(x)=\frac{2}{(x+2)^2}\)

Step 1: Simplify the function

Firstly, simplify the function by dividing both the numerator and the denominator by x, to get \(f(x)=\frac{1}{1+ \frac{2}{x}}\)

Step 2: Differentiate the simplified function

Now, differentiate the simplified function. This can be done by using the chain rule which states that \( (f(g(x)))'=f'(g(x)) \cdot g'(x)\). In our case, f(u) = 1/u, so \(f'(u) = -1 / u^2\). Applying the chain rule gives \(f'(x) = -1/(1+2/x)^2 \cdot (-2/x^2)\).

Step 3: Simplify the derivative

Simplify the derivative by cancelling out the negative signs and simplifying the expression to get \(f'(x)=\frac{2}{x^2(1+ \frac{2}{x})^2}\). Simplifying it further, we get \(f'(x)=\frac{2}{(x+2)^2}\).

Key Concepts

Chain Rule

The Chain Rule is a key concept in differential calculus used when differentiating composite functions, also known as functions within functions. It essentially helps us break down the differentiation process of more complex expressions into simpler parts. Imagine you have a function within another function—this is when the Chain Rule comes in handy.

• Form: If you have a composite function, \( f(g(x)) \), the derivative is given by \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).
• Simplified Approach: Identify the outer function and the inner function. Differentiate each separately.
• Application: Multiply the derivative of the outer function by the derivative of the inner function.

For the given exercise, the outer function is \( f(u) = 1/u \) with its derivative \( f'(u) = -1/u^2 \), and the inner function is \( g(x) = 1 + 2/x \). Keeping these steps in mind makes solving such problems more straightforward.

Quotient Rule

The Quotient Rule is crucial when you have functions of the form \( \frac{u(x)}{v(x)} \), meaning a numerator and a denominator that both depend on \( x \). While typically used for fraction-like functions, some problems can be simplified to avoid the Quotient Rule entirely, as seen in this exercise.

• Formulation: If \( y = \frac{u}{v} \), then \( y' = \frac{v \cdot u' - u \cdot v'}{v^2} \).
• Strategic Alternatives: Simplify such fractions if possible to avoid the Quotient Rule altogether.

In this exercise, simplifying \( f(x) = \frac{x}{x+2} \) first led to a version where the Quotient Rule was unnecessary, saving time and reducing computational errors. Therefore, always review a fraction for simplification before applying the Quotient Rule.

Simplification

Simplification is the process of rewriting expressions to make them easier to work with. In calculus, especially when differentiating or integrating, simplification can save both effort and confusion by reducing complex fractions or expressions into more manageable forms.

• Divide: Break down complex fractions by dividing terms as shown in the given exercise.
• Factor: Look for common factors or ways to combine terms.

In the exercise, the initial function \( f(x) = \frac{x}{x+2} \) was transformed into a simpler form \( f(x) = \frac{1}{1+\frac{2}{x}} \) by dividing through by \( x \). This made it much easier to apply the Chain Rule later on.

Rates of Change

Rates of change describe how variables change with respect to each other. In calculus, derivatives express rates of change. When a function describes a real-world scenario, its derivative tells us how quickly something is happening.

• Interpretation: If \( y = f(x) \), the derivative \( f'(x) \) gives the rate at which \( y \) changes with respect to \( x \).
• Applications: Rates of change are pivotal in fields like physics, economics, and biology where change dynamics are studied.

For the function \( f(x) \), the derivative \( f'(x) = \frac{2}{(x+2)^2} \) provides information about how the output \( f(x) \) changes as \( x \) moves. Understanding rates of change helps us predict and analyze real-world behaviors captured by mathematical models.