Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

\(y' = 2(x + \sin x)(1 + \cos x)\)

Step 1: Identify the Outer Function and Inner Function

The given function is in the form of a composite function, where \(y = (u)^2\) and \(u = x + \sin x\). The outer function is \(g(u) = u^2\) and the inner function is \(u(x) = x + \sin x\).

Step 2: Differentiate the Outer Function

Differentiate the outer function \(g(u) = u^2\) with respect to \(u\) to get \(g'(u) = 2u\).

Step 3: Differentiate the Inner Function

Differentiate the inner function \(u(x) = x + \sin x\) with respect to \(x\). The derivative \(u'(x)\) is calculated as follows: \(\frac{d}{dx}(x) = 1\) and \(\frac{d}{dx}(\sin x) = \cos x\). Therefore, \(u'(x) = 1 + \cos x\).

Step 4: Apply the Chain Rule

The chain rule states that the derivative of \(y = g(u(x))\) with respect to \(x\) is \(y' = g'(u) \cdot u'(x)\). Substituting the derivatives from previous steps, we have: \[y' = 2u \cdot (1 + \cos x)\].

Step 5: Substitute the Expression for \(u\)

We substitute back the expression for \(u\), which is \(u = x + \sin x\). Thus, \[y' = 2(x + \sin x)(1 + \cos x)\].

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used for finding the derivative of a composite function. Understanding it is essential for dealing with functions that nest within others. Here is how you can break it down:

• The chain rule allows us to differentiate composite functions, which are functions made by combining two or more functions.
• It states that if you have a composite function \( y = f(g(x)) \), the derivative \( y' \) is given by \( f'(g(x)) \cdot g'(x) \).
• This means you take the derivative of the outer function and multiply it by the derivative of the inner function.

Think of it as peeling an onion, where each layer represents a different function. You differentiate each layer step-by-step until you reach the core. This rule is critical when we encounter complex functions like \((x + \sin x)^2\).

Composite Function

A composite function is formed when one function is applied to the result of another function. In simpler terms, it's like assembling a sequence of functions where each function feeds into the next.

When dealing with composite functions, you essentially look at:

• An outer function \( g(u) \), for example, \( u^2 \).
• An inner function \( u(x) = x + \sin x \).

Composite functions are everywhere in the real world, from physics equations to economics models. In our example, the expression \( (x + \sin x)^2 \) is a composite function where squaring happens after calculating \( x + \sin x \). Recognizing these components simplifies the differentiation process with the chain rule.

Differentiation

Differentiation is the process of finding the derivative of a function. It provides the rate at which a function is changing at any given point. Through differentiation, you can find slopes of curves, velocities in physics, and more.

Basic rules of differentiation include:

• The derivative of \( x^n \) is \( nx^{n-1} \).
• The derivative of \( \sin x \) is \( \cos x \).
• Use the product, quotient, and chain rules for more complex functions.

Differentiation transforms a curve into a tangible number telling us how steep the curve is at a precise location. In our problem, differentiating the inside and outside parts of \( (x + \sin x)^2 \) illustrates how combined differentiation rules work seamlessly.

Trigonometric Differentiation

Trigonometric differentiation refers to the derivative calculations involving trigonometric functions like \( \sin x \), \( \cos x \), and others. These functions have specific rules for differentiation:

• \( \frac{d}{dx}(\sin x) = \cos x \)
• \( \frac{d}{dx}(\cos x) = -\sin x \)
• Other functions like \( \tan x \) have their own rules derived from these basic identities.

Understanding these rules helps you tackle functions that incorporate angles and periodic behavior. In our exercise, knowing that the derivative of \( \sin x \) is \( \cos x \) was crucial for applying the chain rule efficiently, giving us the rate of change for the whole expression \((x + \sin x)^2\). These derivatives are especially valuable in engineering and physics where wave and oscillation analysis is needed.