Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-20, find \(D_{x} y\). $$ y=\cos ^{3} x $$

Step-by-Step Solution

Answer

\(D_{x} y = -3 \cos^2 x \sin x\).

Step 1: Identify the Problem Type

We need to find the derivative of a trigonometric function raised to a power, specifically \( y = \cos^3 x \). This involves using the chain rule for differentiation.

Step 2: Use the Chain Rule

Recognize that \( y = \cos^3 x \) is equivalent to \( y = (\cos x)^3 \). We will apply the chain rule, which states that \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).

Step 3: Differentiate the Outer Function

Consider \( (\cos x)^3 \) as an outer function \( u^3 \) where \( u = \cos x \). The derivative of \( u^3 \) with respect to \( u \) is \( 3u^2 \).

Step 4: Differentiate the Inner Function

Now find the derivative of the inner function \( u = \cos x \) with respect to \( x \). The derivative of \( \cos x \) is \( -\sin x \).

Step 5: Apply the Chain Rule

Combine the results from Steps 3 and 4 using the chain rule: \[ \frac{dy}{dx} = 3(\cos x)^2 \cdot (-\sin x) \].

Step 6: Simplify the Result

Simplify the expression obtained: \[ \frac{dy}{dx} = -3 \cos^2 x \sin x \].

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus that allows us to differentiate compositions of functions. It is particularly useful when dealing with functions nested within each other, like in our example: \( y = \cos^3 x \). To employ the chain rule, we first identify this as \( y = (\cos x)^3 \), where the outer function is a power function and the inner function is the cosine function.

The rule states that if you have a function \( y = f(g(x)) \), then the derivative is found as \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \). This means we take the derivative of the outer function, leaving the inner function unchanged, and then multiply it by the derivative of the inner function itself.

Applied to our problem, the outer function \( u^3 \) with respect to \( u = \cos x \) provides \( 3u^2 \). Differentiating the inner function \( \cos x \) with respect to \( x \) yields \( -\sin x \).

By applying these derivatives, the chain rule allows us to construct the complete derivative of the original function.

Trigonometric Differentiation

Trigonometric differentiation involves finding derivatives of trigonometric functions such as \( \sin x \), \( \cos x \), \( \tan x \), and others. These functions have specific differentiation rules:

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

In our example, the derivative of \( \cos x \) is \(-\sin x\). This result is critical within the chain rule process. To differentiate functions in which trigonometric functions are compounded or raised to powers, applying these basic derivatives becomes essential.

Understanding the basic trigonometric derivatives lets us handle more complex expressions and use tools like the chain rule effectively.

Calculus Problem Solving

Solving calculus problems often requires breaking them down into manageable steps. By approaching the given problem logically and systematically, we can find the solution efficiently. Let's consider our example, which involves solving for the derivative of \( y = \cos^3 x \).

A structured approach includes:

• Identifying the problem type: Recognizing the expression as a trigonometric function raised to a power.
• Applying appropriate rules: Utilizing the chain rule because of the function’s composition, and trigonometric differentiation.
• Simplifying the result: Incorporating algebraic techniques to arrive at a clear and concise answer.

The result for the derivative, \( \frac{dy}{dx} = -3 \cos^2 x \sin x \), illustrates the power of calculus techniques to solve complex expressions. Each step involves careful consideration and application of foundational concepts, making calculus problem-solving rewarding and insightful for mastering mathematical analysis.