Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(d y\). $$ y=(\tan x+1)^{3} $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 3(\tan x + 1)^2 \sec^2 x \)

Step 1: Identify the function to differentiate

The given function is \( y = ( an x + 1)^3 \). We need to differentiate this function with respect to \( x \).

Step 2: Apply the Chain Rule

To differentiate \( y = ( an x + 1)^3 \), we use the chain rule which states: if \( y = u^n \), then \( \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \). Here, \( u = \tan x + 1 \) and \( n = 3 \).

Step 3: Differentiate the outer function

Using the chain rule, differentiate the outer function \( u^3 \) with respect to \( u \): \( 3 \cdot u^2 \).

Step 4: Differentiate the inner function

Differentiate \( u = \tan x + 1 \) with respect to \( x \): \( \frac{d}{dx}( an x + 1) = \sec^2 x \) since the derivative of \( \tan x \) is \( \sec^2 x \) and the derivative of a constant is zero.

Step 5: Combine results to find \( \frac{dy}{dx} \)

Multiply the derivative of the outer function by the derivative of the inner function: \[ \frac{dy}{dx} = 3 \cdot (\tan x + 1)^2 \cdot \sec^2 x \].

Step 6: Write the final result

Thus, the derivative of \( y = (\tan x + 1)^3 \) with respect to \( x \) is \( \frac{dy}{dx} = 3(\tan x + 1)^2 \sec^2 x \).

Key Concepts

Chain Rule

The chain rule is an essential concept in calculus differentiation, especially when dealing with composite functions. A composite function is a function made by combining two or more functions. Consider it as a function within another function. The chain rule helps us differentiate these composite functions effectively.

Imagine you have a function in the form of \( y = f(u) \), where \( u = g(x) \). To find \( \frac{dy}{dx} \), the chain rule suggests that you differentiate \( f \) with respect to \( u \), then multiply by the derivative of \( u \) with respect to \( x \). In formula terms, this looks like:

• \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

Using this, we can break down the problem into smaller, more manageable parts, making complicated differentiation tasks simpler and systematic. Understanding the chain rule allows you to tackle a wide range of functions, especially when they include layers of operation.

Derivative Calculation

The process of differentiation, or finding a derivative, involves calculating the rate at which a function changes with respect to a variable, typically \( x \). For a given function, its derivative represents the slope of the tangent line to the curve of the function at any point.

For example, if your function is \( y = (\tan x + 1)^3 \), you should begin by identifying its structure: it's a power function with an inner trigonometric function. The derivative calculation entails two main steps:

• Differentiating the outer function: Treat \( (\tan x + 1)^3 \) as \( u^3 \). Its derivative with respect to \( u \) is \( 3u^2 \), according to basic power rule differentiation.
• Differentiating the inner function: The inner function \( u = \tan x + 1 \) translates to \( \frac{du}{dx} = \sec^2 x \), since the derivative of \( \tan x \) is \( \sec^2 x \).

By multiplying these results according to the chain rule, you seamlessly work through the derivative calculation, effectively unpacking the layers of the function to reveal the rate of change.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in both pure and applied mathematics. Differentiating these functions forms a core part of calculus. Each trigonometric function has a distinct derivative which is crucial to remember.

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

Understanding these derivatives allows you to tackle problems involving cyclic patterns and can be especially useful when they occur as parts of larger expressions.For instance, in our problem, differentiating \( \tan x \), a commonly used trigonometric function, results in \( \sec^2 x \). Recognizing and applying these derivatives to trigonometric functions within complex expressions are foundational skills for solving calculus problems efficiently and effectively.