Problem 80 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \begin{equation}y=\log _{7}\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right)\end{equation}

Step-by-Step Solution

Answer

The derivative is \( y' = \frac{1}{\ln(7)} \left( \cot \theta - \tan \theta - 1 - \ln(2) \right) \).

Step 1: Apply the Change of Base Formula for Logarithms

First, we use the change of base formula to express the logarithm with base 7 in terms of natural logarithms (base e). The formula is: \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \). Thus, \( y = \frac{\ln\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right)}{\ln(7)} \).

Step 2: Simplify the Expression Inside the Logarithm

Simplify the expression \( \frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}} \) to make differentiation easier. This can be rewritten as \( \sin \theta \cos \theta \cdot e^{-\theta} \cdot 2^{-\theta} \). Use identities if needed.

Step 3: Use Logarithmic Properties to Simplify Further

Use properties of logarithms to break down the expression: \( \ln\left(\sin \theta \cos \theta \cdot e^{-\theta} \cdot 2^{-\theta}\right) = \ln(\sin \theta) + \ln(\cos \theta) - \theta \ln(e) - \theta \ln(2) \). Simplifying further, it becomes \( \ln(\sin \theta) + \ln(\cos \theta) - \theta - \theta \ln(2) \).

Step 4: Differentiate Using Chain Rule

Differentiate \( y = \frac{\ln(\sin \theta) + \ln(\cos \theta) - \theta - \theta \ln(2)}{\ln(7)} \) with respect to \( \theta \). Derivative of \( \ln(\sin \theta) \) is \( \cot \theta \); of \( \ln(\cos \theta) \) is \( -\tan \theta \); of \( -\theta \) is \( -1 \); and of \( -\theta \ln(2) \) is \( -\ln(2) \).

Step 5: Put It All Together

Combine these results to find the final derivative: \( y' = \frac{1}{\ln(7)} \left( \cot \theta - \tan \theta - 1 - \ln(2) \right) \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a method that simplifies the process of finding derivatives, especially when dealing with complex expressions involving products, quotients, or powers. In this exercise, we aim to find the derivative of a log function with a base other than 10 or the natural base e. The function given in the problem is:

• A logarithmic function with base 7: \( y = \log_{7}\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right) \)

To differentiate this function easily, we first change the base of the logarithm to the natural log using the change of base formula, which allows us to write an arbitrary base logarithm in terms of natural logarithms:

• \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \)
• Thus, our function becomes: \( y = \frac{\ln\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right)}{\ln(7)} \)

Once in this form, we can use differentiation rules for natural logarithms to simplify our calculations.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a triangle to the lengths of its sides. In the given exercise, we are required to differentiate an expression involving sine and cosine trigonometric functions. The expression inside the logarithm is:

• \( \frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}} \)

To manage these trigonometric functions effectively, we simplify the expression by multiplying sine and cosine while introducing their identities to facilitate easier differentiation:

• \( \sin \theta \cos \theta = \frac{1}{2} \sin(2\theta) \) (identity not used here, but good for simplification in similar problems)

In this specific problem, after rewriting and simplifying the logarithm, individual derivatives of the logarithmic forms of sine and cosine must be taken:

• Derivative of \( \ln(\sin \theta) \) is \( \cot \theta \)
• Derivative of \( \ln(\cos \theta) \) is \( -\tan \theta \)

These derivatives help us simplify the differentiation process considerably, helping bridge the connections between logarithmic and trigonometric functions.

Change of Base Formula

The change of base formula is an essential tool in solving problems involving logarithms with any base. Often, we encounter logarithms that are not convenient for direct computation or differentiation. This is particularly the case when dealing with bases other than 10 or e. The change of base formula is:

• \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \)

Using this formula, we transformed the original equation:

• \( \log_7\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right) = \frac{\ln\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right)}{\ln(7)} \)

This transformation facilitates the use of differentiation rules for natural logarithms, thus simplifying the differentiation process significantly.

Chain Rule

The chain rule is a fundamental derivative rule used to find the derivative of composite functions. When a function is made up of other functions, the chain rule allows us to differentiate each part separately and then multiply these derivatives together. In the problem:

• We start with a composite function: the natural log of a quotient involving trigonometric and exponential expressions.

To differentiate it, we apply the chain rule, along with other derivative rules. Key steps include the derivatives of the logarithm, as previously discussed, and the following derivative aspects:

• The derivative of \( -\theta \) is \( -1 \).
• The derivative of \( -\theta \ln(2) \) is \( -\ln(2) \).

Finally, each component's derivatives are combined:

• \( y' = \frac{1}{\ln(7)} \left( \cot \theta - \tan \theta - 1 - \ln(2) \right) \)

The chain rule is vital in allowing us to break down and differentiate the complex expression efficiently.