Problem 32 - Calculus Early Transcendentals

Question

Find \(d y / d x\) $$ y=\sin (\tan 3 x) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 3\sec^2(3x)\cos(\tan(3x)) \).

Step 1: Identify the Composite Function

The given function to differentiate is a composition of functions: \( y = \sin(u) \) where \( u = \tan(3x) \). We will need to apply the chain rule twice, first for the sine function and then for the tangent function within the sine.

Step 2: Differentiate the Outer Function

Differentiate the outside function, \( \sin(u) \), with respect to \( u \). The derivative is \( \cos(u) \). So, \( \frac{d}{du}\sin(u) = \cos(u) \).

Step 3: Differentiate the Inner Function

Now, differentiate the inside function \( u = \tan(3x) \). The derivative of \( \tan(v) \) is \( \sec^2(v) \) and for \( v = 3x \), use the chain rule: \( \frac{d}{dx} \tan(3x) = 3\sec^2(3x) \).

Step 4: Apply the Chain Rule

Combine the derivatives using the chain rule. First, differentiate as \( \frac{dy}{du} = \cos(\tan(3x)) \). Then, multiply by \( \frac{du}{dx} = 3\sec^2(3x) \).Thus, \( \frac{dy}{dx} = \cos(\tan(3x)) \cdot 3\sec^2(3x) \).

Key Concepts

Understanding the Chain Rule

The Chain Rule is a fundamental tool in calculus used for differentiating composite functions. Essentially, it provides a way to calculate the derivative of a function that depends on another function. The core idea revolves around multiplying the derivatives of the outer and inner functions.

When you have a function, say \( y = f(g(x)) \), where \( f \) and \( g \) are functions, the chain rule helps in finding \( \frac{dy}{dx} \). Here's how it works:

• Firstly, differentiate \( f \) with respect to \( g \), giving you \( \frac{df}{dg} \).
• Next, differentiate \( g \) with respect to \( x \), resulting in \( \frac{dg}{dx} \).
• The derivative \( \frac{dy}{dx} \) is then the product of both: \( \frac{df}{dg} \times \frac{dg}{dx} \).

The beauty of the chain rule lies in its simplicity and power, allowing for effective differentiation of complex nested functions without untangling them completely. This makes tackling intricate problems, like those involving multiple layers of functions, manageable.

What Are Composite Functions?

Composite functions occur when one function is inside another, effectively creating layers that you must navigate during differentiation. The major challenge with composite functions is that they require you to break down and deal with each layer individually.

For example, consider the function \( y = \sin(\tan(3x)) \). Here, the function involves three layers:

• First layer: Trigonometric identity \( \sin(u) \), where \( u \) is a placeholder for whatever the input might be.
• Second layer: \( u = \tan(v) \), indicating a function of another variable \( v \).
• Third layer: \( v = 3x \), which adjusts the basic variable \( x \) by a multiplicative factor.

In differentiating composite functions, each layer's derivative interacts with the others. This requires careful application of the chain rule for each transition between layers. Composite functions bring complexity to calculus problems, but once broken down, they follow predictable patterns.

Basics of Trigonometric Functions in Differentiation

Trigonometric functions are a staple in calculus, often forming the building blocks of more complex expressions. When it comes to differentiation, it's crucial to remember the derivatives of basic trigonometric functions, as these underpin more challenging problems.

For the primary trigonometric functions:

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

When differentiating a function like \( y = \sin(\tan(3x)) \), you need to use these derivatives effectively, applying the chain rule between transitions. Each trigonometric function adds a unique layer to the problem, meaning the composite's complexity can grow significantly.

Other advanced trigonometric functions like \( \sec(x) \), \( \csc(x) \), and \( \cot(x) \) have their derivatives, but the methodology remains the same. Identify, differentiate, and stitch the results together using the chain rule to unravel the trigonometric layers effortlessly.