Problem 82 - University Calculus: Early Transcendentals

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

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

Step 1: Rewrite the Logarithmic Expression

To simplify the logarithmic differentiation, we rewrite the expression inside the logarithm. We have \[y = \log_7 \left(\frac{\sin \theta \cos \theta}{e^\theta 2^\theta}\right).\]We can use the logarithm property: \[\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\]to obtain:\[y = \log_7(\sin \theta \cos \theta) - \log_7(e^\theta 2^\theta).\]

Step 2: Use Logarithm Properties

Apply the logarithmic identity, \(\log_b(xy) = \log_b(x) + \log_b(y)\), to split the terms:\[y = \log_7(\sin \theta) + \log_7(\cos \theta) - \log_7(e^\theta) - \log_7(2^\theta).\]Using \(\log_b(x^n) = n\cdot \log_b(x)\), we simplify to:\[y = \log_7(\sin \theta) + \log_7(\cos \theta) - \theta \cdot \log_7(e) - \theta \cdot \log_7(2).\]

Step 3: Find the Derivative of Each Term

Differentiate each term separately with respect to \(\theta\):- \(\frac{d}{d\theta}\left(\log_7(\sin \theta)\right) = \frac{1}{\sin \theta \cdot \ln 7} \cdot \cos \theta \) (using the derivative of \(\log_b(x)\)).- \(\frac{d}{d\theta}\left(\log_7(\cos \theta)\right) = \frac{1}{\cos \theta \cdot \ln 7} \cdot (-\sin \theta)\).- \frac{d}{d\theta}\left(-\theta \cdot \log_7(e)\right) = -\log_7(e).- \frac{d}{d\theta}\left(-\theta \cdot \log_7(2)\right) = -\log_7(2).

Step 4: Combine Derivatives

Now, combine all the derivatives:\[\frac{dy}{d\theta} = \left(\frac{\cos \theta}{\sin \theta \cdot \ln 7}\right) + \left(\frac{-\sin \theta}{\cos \theta \cdot \ln 7}\right) - \log_7(e) - \log_7(2).\]Using the identity \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), the expression becomes:\[\frac{dy}{d\theta} = \frac{\cot \theta}{\ln 7} - \frac{\tan \theta}{\ln 7} - \log_7(e) - \log_7(2).\]

Step 5: Final Step: Simplify the Expression

Simplify the expression by factoring out \(\frac{1}{\ln 7}\):\[\frac{dy}{d\theta} = \frac{1}{\ln 7}(\cot \theta - \tan \theta) - \log_7(e) - \log_7(2).\]

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used when dealing with products, quotients, or powers of functions that are cumbersome to differentiate using standard rules. This method takes advantage of the properties of logarithms to simplify the differentiation process.
The essence is to take the logarithm of both sides of an equation where the variable is present in a complicated expression. This simplifies the expression by transforming it into a sum, difference, or product of simpler terms, which are easier to differentiate. For example, if you have a function of the form: \(y=\log _{7}\left(\frac{\sin \theta \cos \theta}{e^{\theta}2^{\theta}}\right)\), you can rewrite it using logarithm properties as a sum or difference of terms:

• \(\log_7(\frac{x}{y}) = \log_7(x) - \log_7(y)\)
• Use this to break complex fractions into simpler parts.

Once expressed, each term can then be differentiated separately, simplifying the differentiation process significantly.

Trigonometric Functions

Trigonometric functions like \(\sin\theta\) and \(\cos\theta\) play a significant role in many differentiation problems. These functions are periodic, and their derivatives exhibit cyclic behavior.
When differentiating trigonometric functions, remember:

• The derivative of \(\sin\theta\) is \(\cos\theta\)
• The derivative of \(\cos\theta\) is \(-\sin\theta\)

In the exercise, the expression \(\log_7(\sin \theta \cos \theta)\) involves trigonometric terms within logarithmic differentiation. Each trigonometric function can be differentiated using its rule:

• \(\frac{d}{d\theta}(\cos \theta) = -\sin \theta\)
• \(\frac{d}{d\theta}(\sin \theta) = \cos \theta\)

The interaction between trigonometric derivatives and logarithms may initially seem complex, but following the differentiation rules makes handling such problems manageable.

Logarithm Properties

The properties of logarithms simplify expression manipulation and differentiation. They are crucial when expressions are products or quotients of multiple functions.
Key properties used in differentiation include:

• The power rule: \(\log_b(x^n) = n \cdot \log_b(x)\), which allows us to bring the exponent down, making derivatives more straightforward.
• The product rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\), which splits the log of a product into individual terms.
• The quotient rule: \(\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)\), aiding in breaking down complex fractions.

Applying these properties in the original exercise allows the expression to be represented as a sum and difference of simpler logs. Then, each component can be differentiated according to its specific rules until the entire expression is reduced to a straightforward derivative calculation. These properties are vital tools that transform a complex differentiation into a manageable series of steps.