Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=\ln (x / y) ; x=\tan t, y=\sec ^{2} t $$

Step-by-Step Solution

Answer

\( \frac{dw}{dt} = \csc 2t - 2 \tan t \)

Step 1: Identify the Functions

First, identify the functions involved in the problem. The function provided is \( w = \ln (x / y) \) with the substitutions \( x = \tan t \) and \( y = \sec^2 t \). This means that \( w = \ln \left( \frac{\tan t}{\sec^2 t} \right) \).

Step 2: Simplify \( w \)

Rewrite the expression for \( w \) as follows: \( w = \ln(\tan t) - \ln(\sec^2 t) \). This simplification uses the property of logarithms that \( \ln(a/b) = \ln a - \ln b \). Further simplify using the identity \( \sec^2 t = 1 + \tan^2 t \), so the expression remains as \( w = \ln(\tan t) - 2 \ln(\sec t) \).

Step 3: Differentiate \( w \) with respect to \( t \)

Use the chain rule to differentiate each term:- The derivative of \( \ln(\tan t) \) is \( \frac{1}{\tan t} \cdot \sec^2 t \).- The derivative of \( 2 \ln(\sec t) \) is \( 2 \cdot \frac{1}{\sec t} \cdot \sec t \tan t \).

Step 4: Simplify the Derivatives

Calculate each derivative:\( \frac{d}{dt}(\ln(\tan t)) = \frac{\sec^2 t}{\tan t} \)\( \frac{d}{dt}(2 \ln(\sec t)) = 2 \tan t \)

Step 5: Combine the Derivatives

Combine the derivatives to find \( \frac{dw}{dt} \):\( \frac{dw}{dt} = \frac{\sec^2 t}{\tan t} - 2 \tan t \)

Step 6: Final Simplification

The expression can be simplified as:\( \frac{dw}{dt} = \frac{1}{\sin t \cos t} - 2 \tan t \)This uses the identity \( \sec^2 t = \frac{1}{\cos^2 t} \) and \( \tan t = \frac{\sin t}{\cos t} \). So the equivalent simplified derivative is \( \frac{dw}{dt} = \csc 2t - 2 \tan t \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a method used to simplify the process of differentiating complex functions. It is especially useful when dealing with products or quotients within a function. The basic idea is to take the natural logarithm of both sides of an equation, which allows us to utilize properties of logarithms to make differentiation more manageable. In this exercise, we used logarithmic differentiation for the function

• \( w = \ln \left( \frac{x}{y} \right) \)
• by recognizing that it can be rewritten as \( \ln(x) - \ln(y) \).

This simplifies the differentiation process because the derivatives of logarithmic functions, such as \( \ln(x) \), are straightforward to handle.
Logarithmic differentiation leverages properties like:

• \( \ln(a \cdot b) = \ln a + \ln b \)
• \( \ln \left(\frac{a}{b}\right) = \ln a - \ln b \)

This approach can drastically simplify complex functions and ease the differentiation process.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying expressions during differentiation. These identities are formulas involving trigonometric functions, which can transform or simplify expressions. In this problem, we use identities to simplify the expressions for functions like \( \tan t \) and \( \sec t \):

• \( \sec^2 t = 1 + \tan^2 t \)
• \( \tan t = \frac{\sin t}{\cos t} \)
• \( \sec t = \frac{1}{\cos t} \)

These identities help us rewrite and reformulate the function \( w = \ln(\tan t) - 2 \ln(\sec t) \). Without using these identities, differentiating \( \ln \left( \frac{\tan t}{\sec^2 t} \right) \) directly would be much more challenging. By expressing terms in their simplest forms, we make the differentiation process straightforward and more efficient.

Differentiation Techniques

Differentiation is a fundamental concept in calculus involving numerous techniques to find derivatives effectively. Techniques include using the product rule, quotient rule, and the chain rule.

In particular, the chain rule is pivotal in this exercise. It is utilized whenever a function is composed of other functions, as in the case of \( w = \ln(\tan t) - 2 \ln(\sec t) \). The chain rule states that to differentiate a composite function, you must differentiate the outer function and multiply it by the derivative of the inner function.

• For \( \ln(\tan t) \), differentiate as \( \frac{1}{\tan t} \cdot \sec^2 t \).
• For \( 2 \ln(\sec t) \), differentiate as \( 2 \cdot \frac{1}{\sec t} \cdot \sec t \tan t \) resulting in \( 2 \tan t \).

This structured approach using differentiation techniques allows for accurate and efficient calculation of derivatives, even for complex expressions.