Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated derivative. \(y^{\prime}\) where \(y=\left(x^{2}+4\right)^{2}\)

Step-by-Step Solution

Answer

The derivative \(y'\) is \(4x^3 + 16x\).

Step 1: Identify the Function Type

The given function is \(y = (x^2 + 4)^2\). It is a composite function, often recognized as an outer function \((u(x))^n\) where \(u(x) = x^2 + 4\), and the inner function \(x^2 + 4\) itself.

Step 2: Apply the Chain Rule

To differentiate a composite function, we use the chain rule, which states \(\frac{d}{dx} (u(x))^n = n(u(x))^{n-1} \cdot u'(x)\). Here, \(n = 2\) and \(u(x) = x^2 + 4\).

Step 3: Differentiate the Outer Function

Differentiate the outer part \((u(x))^2\) with respect to \(u\). This gives \(2(u)^{1}\). Substituting \(u\) gives \(2(x^2 + 4)\).

Step 4: Differentiate the Inner Function

Now differentiate the inner function \(u(x) = x^2 + 4\) with respect to \(x\). This results in \(u'(x) = 2x\).

Step 5: Combine Results Using the Chain Rule

Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function: \(2(x^2 + 4) \cdot 2x\). This simplifies to \(4x(x^2 + 4)\).

Step 6: Simplify the Result

Distribute \(4x\) inside the parenthesis: \(4x(x^2 + 4) = 4x^3 + 16x\). So, \(y' = 4x^3 + 16x\).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in differentiation that allows us to handle composite functions. It's like a rule book for differentiating complex expressions. Think of it as a process of peeling layers, where you first focus on the larger structure and then work inward. The chain rule states:

• For any composite function in the form \[ y = (u(x))^n \] its derivative is calculated by the formula \[ \frac{d}{dx} (u(x))^n = n(u(x))^{n-1} \cdot u'(x) \].

Here, you differentiate the outer function first, then multiply it by the derivative of what's inside. It's crucial when you have functions wrapped within other functions, like our current exercise. So, for \( y = (x^2 + 4)^2 \), the outer function \((u(x))^n\)is first differentiated to get \( 2(x^2 + 4)^1 \),and then the inner function \( x^2 + 4 \) is differentiated into \( 2x \). Apply the chain rule by multiplying these two results, and you unlock the derivative.

Composite Function

A composite function is like a mathematical nested doll, where one function sits inside another. In simpler terms, it's a function where one function's output becomes another function’s input. Mathematically, you can write a composite function as \( f(g(x)) \), where \( f \) and \( g \) are individual functions.In the given exercise, \((x^2 + 4)^2 \) is our composite function. Here, \( x^2 + 4 \) serves as the inner function, while raising it to the power of 2 forms the outer function. Recognizing this composite nature is essential before applying the chain rule. Composite functions are particularly common in calculus, as real-world scenarios often involve multiple interconnected processes. Understanding how to break them down simplifies differentiation. For example, knowing how to decouple the inner and outer layers allows you to directly apply techniques like the chain rule, streamlining the problem-solving process.

Differentiation

Differentiation involves finding the rate at which a function changes at a given point. Think of it as calculating how steep a slope is at any spot on a curve. This process is key in calculus, as it helps us understand how quantities evolve over time or space. When you differentiate, you're essentially identifying how a small change in the input (like \( x \)) results in a change in the output (like \( y \)). In the context of our exercise, we started with \( y = (x^2 + 4)^2 \).Through differentiation, we calculated that \( y' = 4x^3 + 16x \), which reveals how the function's slope varies with \( x \). Different techniques, like the chain rule, product rule, or quotient rule, help navigate differentiation’s complexities. In some cases, functions will be straightforward, while others might require carefully untangling nested operations. Mastery of differentiation opens doors to understanding motion, optimizing systems, and solving real-world dynamic problems.

Calculus Problem-Solving

Solving calculus problems is akin to solving a puzzle, where each piece needs to be placed just right. It involves following structured steps while interpreting mathematical concepts. Whether you're tackling a straightforward derivative or a complex integral, the problem-solving process follows specific paths. To solve problems like our current derivative exercise, you:

• Begin by identifying the function type, knowing if it's composite, polynomial, or trigonometric.
• Apply appropriate rules, like the chain rule for composite functions.
• Calculate derivatives step by step, moving from outer layers to inner functions.
• Simplify your result to ensure it's as clean and interpretable as possible.

Effective calculus problem-solving doesn't just follow recipes; it also requires an understanding of concepts and techniques, with room for intuition and creativity. In real-world scenarios, being adept at these calculations allows scientists, engineers, and economists to create better models, simulate outcomes, and derive valuable insights about the world.