Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-18, find \(D_{x} y\). $$ y=\sin ^{2} x $$

Step-by-Step Solution

Answer

The derivative is \( D_x y = \sin 2x \).

Step 1: Identify the Function

The given function is \[ y = \sin^2 x \]This function is composed of the sine function \( \sin x \) squared.

Step 2: Apply Chain Rule

To find \( D_{x} y \), we need to use the chain rule. The chain rule for finding the derivative of \( (f(x))^n \) is\[ \frac{d}{dx}(f(x))^n = n(f(x))^{n-1} \cdot f'(x) \]In this case, \( f(x) = \sin x \) and \( n = 2 \).

Step 3: Derive the Outer Function

First, differentiate the outer function \( u = x^2 \) with respect to \( u \) itself where \( u = \sin x \): \[ \frac{d}{du} u^2 = 2u \]Thus, the derivative is \[ 2\sin x \]

Step 4: Derive the Inner Function

Now differentiate the inner function \( \sin x \) with respect to \( x \):\[ \frac{d}{dx} \sin x = \cos x \]

Step 5: Combine Using the Chain Rule

Combine the derivatives of the outer and inner functions using the chain rule:\[ D_{x} y = 2\sin x \cdot \cos x \]

Step 6: Express in Simplified Form

The expression can be written using the identity for the double angle:\[ \sin 2x = 2\sin x \cos x \]Therefore, the derivative is:\[ D_{x} y = \sin 2x \]

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. When you have a function inside another function, like \( (f(x))^n \), the chain rule helps you find the derivative. Think of it like peeling an onion: you tackle the outside layer first, then the inside.
This rule is crucial when dealing with nested functions because it allows us to break down the complicated function into simpler parts. The general formula is:

• \( \frac{d}{dx}(f(x))^n = n(f(x))^{n-1} \cdot f'(x) \)

You first differentiate the outer function, then the inner one. For example, with \( y = \sin^2 x \), treat \( \sin x \) as the inner function and squaring as the outer.
This method allows you to smoothly connect the derivative of the outer function to the derivative of the inner function, giving you the overall derivative of the composite function.

Trigonometric Functions

Trigonometric functions like \( \sin x \), \( \cos x \), and \( \tan x \) are foundational in calculus due to their periodic behavior and derivatives. When differentiating functions that involve trigonometric expressions, it's important to know their specific derivatives:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).

These derivatives help in solving various differentiation problems featuring trigonometric functions. In our example, determining \( \frac{d}{dx} \sin x = \cos x \) was essential to finding the derivative of \( y = \sin^2 x \).
Recognizing these patterns and identities, like the double angle formula \( \sin 2x = 2\sin x \cos x \), speeds up your problem-solving process when working with derivatives involving trigonometric functions.

Differentiation Techniques

Differentiation is a primary tool in calculus for determining how a function changes. It’s the process of finding a derivative, which tells us the rate of change at any given point. Numerous techniques exist for taking derivatives:

• Direct differentiation: Used for basic functions where we apply known derivative rules.
• Chain rule: Essential for differentiating composition of functions, as discussed in detail.
• Product and quotient rule: Used when dealing with multiplication or division of two functions.

Using these methods effectively allows us to handle more complex problems easily. In the example problem, we utilized the chain rule as a part of our toolkit.
This technique allowed us to differentiate a complex function step-by-step methodically, demonstrating how using differentiation techniques strategically can simplify various tasks in calculus. Understanding these methods and knowing when to apply them is critical for solving real-world problems that involve change.