Problem 235 - Calculus Volume 1

Question

For the following exercises, find \(\frac{d y}{d x}\) for each function. $$y=\sin (\cos 7 x)$$

Step-by-Step Solution

Answer

The derivative is \(-7\sin(7x)\cos(\cos(7x))\)."

Step 1: Identify the Structure

Recognize that we have a composition of two trigonometric functions. The outer function is \( y = \sin(u) \) where \( u = \cos(7x) \). This tells us that the Chain Rule will be needed.

Step 2: Apply the Chain Rule

The Chain Rule states that \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \). Our goal is to find these two derivatives and multiply them together.

Step 3: Differentiate the Outer Function

Determine \( \frac{dy}{du} \) where \( y = \sin(u) \). The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \). Thus, \( \frac{dy}{du} = \cos(u) = \cos(\cos(7x)) \).

Step 4: Differentiate the Inner Function

Determine \( \frac{du}{dx} \) where \( u = \cos(7x) \). The derivative of \( \cos(7x) \) with respect to \( x \) is \( -7\sin(7x) \), since \( \frac{d}{dx}\cos(kx) = -k\sin(kx) \) for a constant \( k \).

Step 5: Combine Using the Chain Rule

Substitute back into the Chain Rule: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(\cos(7x)) \times (-7\sin(7x)) \]Thus, the derivative is \(-7\sin(7x)\cos(\cos(7x))\).

Key Concepts

Chain Rule

The concept of the Chain Rule in calculus is a powerful method for finding the derivative of composite functions. When you have a function nested within another function, the Chain Rule allows you to differentiate it efficiently. It's like peeling layers from an onion - you tackle the outer layer first, then move inwards.

To apply the Chain Rule, let's remember this essential formula: for a composite function, say \( y = f(g(x)) \), the derivative \( \frac{dy}{dx} \) is found by multiplying the derivative of the outer function \( \frac{dy}{du} \) with respect to \( u \), by the derivative of the inner function \( \frac{du}{dx} \).

## Why Use the Chain Rule?

• It simplifies differentiating complicated functions by breaking them into simpler parts.
• It's crucial for working with functions expressed in a form where one function is inside another.

In our exercise, the outer function was \( y = \sin(u) \) and the inner function was \( u = \cos(7x) \). By following the Chain Rule, we successively found the derivatives and then combined them to solve the problem.

Trigonometric Differentiation

Trigonometric differentiation involves finding the derivatives of trigonometric functions. These functions include \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), and others. Each has a different rule for differentiation, which is crucial to solving calculus problems involving trigonometric expressions.

### Basic Derivative 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 specific exercise, we needed the derivative of \( \cos(7x) \), which is a bit trickier because of the coefficient before \( x \). Using the chain rule context, it calculated to \( -7\sin(7x) \), adjusting for the factor from the outside derivative.

Mastering these differentiation rules allows us to tackle complex functions that include trigonometric components, like the one in the exercise with \( \sin(\cos(7x)) \).

Composite Functions

Composite functions consist of one function nested inside another. Understanding how to differentiate them is essential in calculus as it appears frequently in problems involving varying variables.

### Breaking Down Composite Functions

• Identify the inner and outer functions.
• Apply derivatives step-by-step using the Chain Rule.
• Find each derivative and multiply them together to get the final result.

In the exercise, \( y = \sin(\cos(7x)) \) is a composite function because \( \sin \) is applied to \( \cos(7x) \). Recognizing this structure is the first step in applying the Chain Rule.

Once you've practiced identifying and differentiating composite functions, problems like our example become straightforward: identify, differentiate, then simplify to reach your answer.