Problem 11 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=\left(x^{2}+1\right)^{3} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = (x^2 + 1)^3 \) is \( 6x(x^2 + 1)^2 \).

Step 1: Identify the Composition Function

The function is given as \( f(x) = (x^2 + 1)^3 \). This is a composition of the outer function \( u^3 \) and the inner function \( u = x^2 + 1 \).

Step 2: Apply the Generalized Power Rule

Using the generalized power rule \( \frac{d}{dx}[u^n] = n \, u^{n-1} \, \frac{du}{dx} \), identify \( n = 3 \) and \( u = x^2 + 1 \).

Step 3: Differentiate the Inner Function

Find the derivative of \( u = x^2 + 1 \). The derivative, \( \frac{du}{dx} = 2x \).

Step 4: Derive Using the Chain Rule

Apply the chain rule: \( \frac{d}{dx}[(x^2 + 1)^3] = 3 \cdot (x^2 + 1)^{3-1} \cdot \frac{d}{dx}(x^2 + 1) \).

Step 5: Complete the Derivative Calculation

Substitute \( \frac{du}{dx} = 2x \) into the equation: \( 3 \cdot (x^2 + 1)^{2} \cdot 2x = 6x(x^2 + 1)^{2} \). This is the derivative of \( f(x) \).

Key Concepts

Chain Rule

The chain rule is a formula used to find the derivative of composite functions. A composite function, as we'll discuss, involves one function inside another. In essence, the chain rule helps you "chain" derivatives together in a sequence.

When applying the chain rule, you differentiate the outer function and then multiply it by the derivative of the inner function. This method is crucial for differentiating functions where one function is nested within another. It simplifies the process, making it possible to handle even the most complex compositions.

For example, in the function \(f(x) = (x^2 + 1)^3\), the chain rule allows us to differentiate smoothly by dealing first with the power, then the base function. The goal is to link them correctly using their derivatives.

Composition Function

A composition function involves two or more functions combined into one. Think of it as a function formed by substituting another function into it. It is fundamental because it transforms how complex equations are solved in calculus.

Taking the exercise's function \(f(x) = (x^2 + 1)^3\), we recognize this as a composition of:

• Inner function: \(u = x^2 + 1\)
• Outer function: \(u^3\)

The outer function takes on the role of applying a power to what's inside, while the inner function retains the core variable dependency. This hierarchical structure of functions within functions underpins many aspects of calculus and problem-solving.

Calculus Derivative

In calculus, a derivative measures how a function changes as its input changes—in simpler terms, how much a specific function "moves" or "slopes." Derivatives are the backbone of understanding changes and rates in a multitude of scientific and engineering applications.

For the function from the exercise, finding the derivative involves determining how \(f(x) = (x^2 + 1)^3\) changes. By using rules such as the generalized power rule and chain rule, we extract the expression \(6x(x^2 + 1)^2\). This represents the rate of change for the function across different values of \(x\).

Derivatives give insight into how functions behave, predict trajectories, optimize solutions, and solve physical problems, making them essential in calculus.