Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The derivative is \( 2\tan(x)\sec^2(x) \).

Step 1: Identify the Function

The function given is a composition: \( y = \tan^2(x) \). You can rewrite it as \( y = (\tan(x))^2 \). This will help us in applying the chain rule.

Step 2: Declare the Outer and Inner Functions

For applying the chain rule, identify the outer function \( u^2 \), where \( u = \tan(x) \), and the inner function \( \tan(x) \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( u^2 \) with respect to \( u \), which gives \( \frac{d}{du}(u^2) = 2u \). Here, \( u = \tan(x) \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( \tan(x) \) with respect to \( x \), which gives \( \frac{d}{dx}(\tan(x)) = \sec^2(x) \).

Step 5: Apply the Chain Rule

According to the chain rule, the derivative of \( y = (\tan(x))^2 \) can be found by multiplying the derivatives of the outer and inner functions: \( D_x y = \frac{dy}{du} \cdot \frac{du}{dx} = 2\tan(x) \cdot \sec^2(x) \).

Step 6: Simplify the Expression

The derivative \( D_x y \) is \( 2\tan(x)\sec^2(x) \). This is the simplified form of the derivative of \( y = \tan^2(x) \).

Key Concepts

Chain Rule

When we talk about finding the derivative of a composite function like \( y = \tan^2(x) \), the chain rule is our friend. It helps us differentiate functions that are nested within each other, breaking them into more manageable parts. Think of the chain rule as a two-step process:

• Identify the parts: First, recognize the outer and inner functions. In this case, the outer function is \( u^2 \) and the inner function is \( u = \tan(x) \).
• Differentiate the components: Take the derivative of the outer function with respect to the inner function. Then take the derivative of the inner function with respect to \( x \). The chain rule neatly ties these pieces together through multiplication.

By applying the chain rule, you'll be able to tackle complex derivatives more confidently. Notice how it enables us to break down what seems complicated into a step-by-step processes.

Trigonometric Functions

Trigonometric functions like \( \tan(x) \), \( \sin(x) \), and \( \cos(x) \) go hand in hand with calculus, especially when finding derivatives. Each of these functions has a specific rule for differentiation:

• \( \frac{d}{dx}(\sin(x)) = \cos(x) \)
• \( \frac{d}{dx}(\cos(x)) = -\sin(x) \)
• \( \frac{d}{dx}(\tan(x)) = \sec^2(x) \)

In our example, \( y = \tan^2(x) \), we focus on \( \tan(x) \) and its derivative, \( \sec^2(x) \). Knowing these derivatives is crucial. It allows us to apply the chain rule and solve the problem correctly. These trigonometric derivatives are among the most common in calculus, so it's good to remember them for quick reference.

Differentiation Steps

Differentiation is essentially the method we use to find the rate of change of a function. In our problem, determining \( D_x y \) involves several steps, which, when followed systematically, make the task straightforward. Here’s a breakdown:

• Identify the Function: Recognize that \( y = \tan^2(x) \) is a composite function and rewrite it if necessary.
• Clarify Inner and Outer Functions: Clearly state what constitutes the inner (\( \tan(x) \)) and outer (\( u^2 \)) functions.
• Differentiate the Parts: First find \( \frac{d}{du}(u^2) \) as \( 2u \) and then \( \frac{d}{dx}(\tan(x)) \) as \( \sec^2(x) \).
• Apply the Chain Rule: Multiply these derivatives to obtain the complete derivative \( 2\tan(x)\sec^2(x) \).
• Simplify: Make sure the final expression is as simple as possible for ease of use and understanding.

By following these steps, you can handle derivatives of many composite functions efficiently. The process aids clarity and structure in problem-solving.