Problem 76 - Thomas Calculus in SI Units

Question

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

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{d\theta} = \frac{\cot \theta}{\ln(7)} - \frac{\tan \theta}{\ln(7)} - \log_7(e) \cdot (1 + \ln(2)) \).

Step 1: Simplify the Expression Inside the Logarithm

First, simplify the expression inside the logarithm: \( \frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}} \) can be rewritten as \( \sin \theta \cos \theta / (e^{\theta} \cdot 2^{\theta}) = \frac{\sin \theta \cos \theta}{e^{\theta+ \theta \ln(2)}} \).

Step 2: Use Logarithmic Properties

Apply the properties of logarithms to simplify \( y = \log_7 \left( \frac{\sin \theta \cos \theta}{e^{\theta+ \theta \ln(2)}} \right) \). This can be written as: \[y = \log_7(\sin \theta \cos \theta) - \log_7(e^{\theta+ \theta \ln(2)})\]Further simplification gives:\[y = \log_7(\sin \theta) + \log_7(\cos \theta) - (\theta + \theta \ln(2)) \cdot \log_7(e)\].

Step 3: Differentiate Each Term with Respect to \(\theta\)

Differentiate each term separately using the chain rule and derivative formulas:- \( \frac{d}{d\theta}[\log_7(\sin \theta)] = \frac{1}{\ln(7) \sin \theta} \cdot \cos \theta = \frac{\cot \theta}{\ln(7)} \).- \( \frac{d}{d\theta}[\log_7(\cos \theta)] = \frac{1}{\ln(7) \cos \theta} \cdot (-\sin \theta) = -\frac{\tan \theta}{\ln(7)} \).- \( \frac{d}{d\theta}[(\theta + \theta \ln(2)) \log_7(e)] = \log_7(e) \cdot (1 + \ln(2)) \).

Step 4: Combine the Derivatives

Put together the derivatives from Step 3:\[\frac{dy}{d\theta} = \frac{\cot \theta}{\ln(7)} - \frac{\tan \theta}{\ln(7)} - \log_7(e) \cdot (1 + \ln(2))\].

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique when dealing with the differentiation of complex functions, particularly when products, quotients, or powers are involved. In this exercise, we used a logarithm with base 7. To begin, we simplified the logarithmic argument, showing it as a quotient of trigonometric functions over exponential ones.
Applying logarithmic properties helps us split the function into simpler components, transforming multiplication and division into addition and subtraction of logs.
In our specific solution, this simplification turned:

• Product inside the log into a sum of logs: \( \log_7(\sin \theta \cos \theta)\) into \(\log_7(\sin \theta) + \log_7(\cos \theta)\).
• Quotient into subtraction: \( \frac{\text{numerator}}{\text{denominator}}\) into difference in logs.
• Handled the exponential function by using properties of exponentials: \( e^{x} = e \cdot x \), reflecting it as linear.

This simplification not only makes differentiation manageable but also highlights the inherent relationships between logarithms and exponentials.

Trigonometric Derivatives

When differentiating terms that involve trigonometric functions, it is crucial to know the basic derivatives of these functions. In our example, differentiation of \(\log_7(\sin \theta)\) and \(\log_7(\cos \theta)\) led to terms involving derivatives of sine and cosine.
By using the formula for the derivative of a log function, combined with knowledge of trigonometric derivatives, we derive:

• The derivative of \(\log_7(\sin \theta)\) results in \( \frac{\cot \theta}{\ln(7)}\).
• The derivative of \(\log_7(\cos \theta)\) gives us \(-\frac{\tan \theta}{\ln(7)}\).

These are achieved using:

• Derivative of \(\sin \theta\): \( \cos \theta\).
• Derivative of \(\cos \theta\): \(-\sin \theta\).

Noticing patterns such as sine turning into cosine and vice versa, simplifies the work, offering a straightforward approach to handling trigonometric derivatives.

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions. It enables us to differentiate functions that fit within larger, more complex functions.
In our situation, the chain rule was necessary for differentiating the logs of the sine and cosine functions. Both involve embedded functions that demand these additional steps:

• For \(\log_7(\sin \theta)\): Differentiate the outer function, log, first, yielding \(\frac{1}{\ln(7) \sin \theta}\), then multiply by the derivative of the inner function, \(\cos \theta\).
• For \(\log_7(\cos \theta)\): Similar steps lead to a result including \(-\sin \theta\).

Recognizing where the chain rule applies is crucial. It ensures that we successfully navigate the layers of a problem, avoiding mistakes that occur when functions are tightly linked or compounded.