Problem 20 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\). \(f(x)=\left(x^{2}+1\right)^{4}\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 8x(x^2 + 1)^3 \).

Step 1: Identify the Outer and Inner Functions

The function is given as \( f(x) = (x^2 + 1)^4 \). Observe that this is a composite function where the outer function is \((u)^4\) and the inner function is \(u = x^2 + 1\).

Step 2: Differentiate the Outer Function

Using the chain rule, we start by differentiating the outer function \( (u)^4 \) with respect to \(u\). The derivative of \(u^4\) with respect to \(u\) is \(4u^3\). This means the derivative of the outer function is \(4(x^2+1)^3\).

Step 3: Differentiate the Inner Function

Next, differentiate the inner function \(u = x^2 + 1\) with respect to \(x\). The derivative of \(x^2+1\) is \(2x\).

Step 4: Apply the Chain Rule

Multiply the derivative of the outer function by the derivative of the inner function: \(f^{\prime}(x) = 4(x^2 + 1)^3 \cdot 2x\).

Step 5: Simplify the Expression

Simplify the expression to obtain the final derivative: \(f^{\prime}(x) = 8x(x^2 + 1)^3\).

Key Concepts

Understanding the Chain Rule

The chain rule is an essential concept in calculus, particularly when dealing with composite functions. It helps us find the derivative of a function that is composed of two or more functions. To apply the chain rule, you follow this fundamental step:

• Differentiating the outer function
• Multiplying by the derivative of the inner function

In our exercise, the function \( f(x) = (x^2 + 1)^4 \) is composed of an inner function and an outer function. By recognizing and differentiating each part properly, you can effectively use the chain rule and calculate the derivative. This rule is especially helpful when you have nested functions, where one function is inside another. Essentially, it is a way to "chain" together the operations of differentiation.

Recognizing and Working with Composite Functions

Composite functions are functions that consist of one function nested within another. In the provided exercise, the expression \( f(x) = (x^2 + 1)^4 \) is a classic example. Here, the inner function is \( u = x^2 + 1 \) and the outer function is \( u^4 \).
To differentiate such a function, you must first separate the inner from the outer function. This is vital because it allows you to manage each part independently before finally combining them using the chain rule. It’s like peeling apart the layers of an onion—handle each layer carefully to understand the whole structure.Knowing how to identify and manipulate these functions forms a foundation for tackling numerous problems in calculus. Once you've practiced this skill with various functions, applying the derivatives becomes an intuitive process.

Derivative Calculation Simplified

Calculating derivatives might seem daunting at first, but breaking it down into smaller tasks can make it manageable. In this problem, to compute \( f^{\prime}(x) \), the chain rule guides us through each step succinctly.

The general process is:

• Differentiate the outer function: For\((u)^4\), this yields \( 4(u)^3 \).
• Differentiate the inner function \( x^2 + 1 \), which gives \( 2x \).
• Finally, multiply these results to form the derivative of the composite function: \( 4(x^2 + 1)^3 \times 2x = 8x(x^2 + 1)^3 \).

Breaking the problem into these steps allows us to approach it methodically. This structured approach to derivative calculation is invaluable for tackling not only this exercise but also a wide variety of calculus problems.