Problem 81 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{d\theta} = \sin(\log_{7} \theta) + \frac{\cos(\log_{7} \theta)}{\ln(7)} \).

Step 1: Identify the Structure

The function given is a product of two functions, \( y = \theta \cdot \sin(\log_{7} \theta) \). We will use the product rule for differentiation, which states \( u'v + uv' \) if \( y = uv \). Let \( u = \theta \) and \( v = \sin(\log_{7} \theta) \).

Step 2: Differentiate First Function

Differentiate \( u = \theta \) with respect to \( \theta \). Since \( u = \theta \), \( u' = \frac{d}{d\theta}(\theta) = 1 \).

Step 3: Differentiate Second Function

Now differentiate \( v = \sin(\log_{7} \theta) \). Use the chain rule: \( v' = \cos(\log_{7} \theta) \cdot \frac{d}{d\theta}(\log_{7} \theta) \). Recall that \( \frac{d}{d\theta}(\log_{a} \theta) = \frac{1}{\theta \ln(a)} \), hence \( v' = \cos(\log_{7} \theta) \cdot \frac{1}{\theta \ln(7)} \).

Step 4: Apply the Product Rule

According to the product rule, the derivative \( \frac{dy}{d\theta} \) is given by \( u'v + uv' \). Substituting the results from the previous steps:\[ \frac{dy}{d\theta} = (1)\sin(\log_{7} \theta) + \theta \cdot \left(\cos(\log_{7} \theta) \cdot \frac{1}{\theta \ln(7)}\right) \] Simplify the expression:\[ \frac{dy}{d\theta} = \sin(\log_{7} \theta) + \frac{\cos(\log_{7} \theta)}{\ln(7)} \]

Key Concepts

Product Rule

The product rule is a fundamental concept within differential calculus that is used when differentiating products of two functions. When you have a function that is the product of two separate functions, say \( y = uv \), the product rule tells us how to find its derivative. It is expressed as \( u'v + uv' \), where \( u \) and \( v \) are functions of the same variable, and \( u' \) and \( v' \) are their respective derivatives.

In the original exercise, the function \( y = \theta \sin(\log_{7} \theta) \) is the product of \( \theta \) and \( \sin(\log_{7} \theta) \). By identifying this structure, we decide that \( u = \theta \) and \( v = \sin(\log_{7} \theta) \).

• First, the derivative of \( u = \theta \) is straightforward: \( u' = 1 \).
• Next, you differentiate \( v = \sin(\log_{7} \theta) \) using further differentiation rules which will be discussed in subsequent sections.
• Finally, substitute these derivatives into the product rule formula to get the derivative of the entire function.

The key point here is to correctly identify when you have a product of two functions, which allows for the proper application of the product rule in finding the derivative.

Chain Rule

The chain rule is essential when differentiating composite functions. A composite function is a function made up of two or more functions. In essence, the chain rule helps find the derivative of a function that is nested within another. Mathematically, if you have a function of \( g(f(x)) \), then its derivative can be found using the chain rule as \( g'(f(x)) \cdot f'(x) \).

For the exercise at hand, we encountered a composite function when differentiating \( \sin(\log_{7} \theta) \). Here, \( \log_{7} \theta \) is the inner function and \( \sin \) is the outer function.

• First, you take the derivative of the outer function, \( \sin \), which becomes \( \cos \).
• Then multiply it by the derivative of the inner function, \( \log_{7} \theta \).
• Using logarithmic differentiation, we find that the derivative of \( \log_{7} \theta \) is \( \frac{1}{\theta \ln(7)} \).

By following these steps, you carefully separate the components of your composite functions, making sure to multiply the outer derivative by the inner one, precisely as dictated by the chain rule.

Derivative of Logarithmic Functions

Logarithmic functions appear often in calculus and have specific rules for differentiation. When taking the derivative of a logarithmic function, such as \( \log_{a} x \), you use the formula \( \frac{1}{x \ln(a)} \). This is crucial when the base of the logarithm is not the natural base \( e \).

In the given problem, you have \( \log_{7} \theta \) as part of the function that was differentiated using the chain rule. The derivative process involved involves:

• Using the specific formula: \( \frac{d}{d\theta}(\log_{7} \theta) = \frac{1}{\theta \ln(7)} \).
• Substituting this derivative back into the equation when applying the chain rule to the sine function.
• Understanding that the natural log \( \ln(a) \) adjusts the rate of change specified by the base of the logarithm.

Understanding how to differentiate logarithmic functions allows you to tackle more complex calculus problems that involve log expressions as part of their functions.