Problem 75 - Thomas Calculus in SI Units

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: Understand the Function

The function we are given is \( y = \theta \sin (\log_{7} \theta) \). Here, \( \theta \) is the independent variable, and \( \log_{7} \theta \) is the logarithm of \( \theta \) to the base 7.

Step 2: Apply Product Rule

The function \( y = \theta \sin (\log_{7} \theta) \) is a product of two functions: \( u = \theta \) and \( v = \sin(\log_{7}\theta) \). According to the product rule, the derivative of the product \( uv \) is \( u'v + uv' \).

Step 3: Differentiate \( u \)

The function \( u = \theta \) is straightforward. The derivative \( u' = \frac{d}{d\theta} (\theta) = 1 \).

Step 4: Differentiate \( v \) using Chain Rule

To find \( v' \), first note \( v = \sin(\log_{7}\theta) \). Let \( g = \log_{7}\theta \), thus \( v = \sin(g) \), and apply the chain rule: 1. Differentiate \( \sin(g) \) with respect to \( g \): \( \cos(g) \). 2. Differentiate \( g = \log_{7}\theta \) with respect to \( \theta \). Using change-of-base formula, \( \log_{7}\theta = \frac{\ln \theta}{\ln 7} \), giving \( \frac{d}{d\theta}(\log_{7}\theta) = \frac{1}{\theta \ln 7} \).Therefore, \( v' = \cos(\log_{7}\theta) \cdot \frac{1}{\theta \ln 7} \).

Step 5: Apply Results in Product Rule Formula

Substitute \( u = \theta \), \( v = \sin(\log_{7}\theta) \), \( u' = 1 \), and \( v' = \cos(\log_{7}\theta) \cdot \frac{1}{\theta \ln 7} \) into the product rule: \[ \frac{dy}{d\theta} = 1 \cdot \sin(\log_{7}\theta) + \theta \cdot \cos(\log_{7}\theta) \cdot \frac{1}{\theta \ln 7} \].

Step 6: Simplify the Derivative Expression

Simplify the expression: \[ \frac{dy}{d\theta} = \sin(\log_{7}\theta) + \frac{\cos(\log_{7}\theta)}{\ln 7} \]. This is the derivative of the original function \( y \) with respect to \( \theta \).

Key Concepts

Understanding the Product Rule

The product rule is a useful technique for finding the derivative of a function composed of two multiplied sub-functions.
For example, if you have a function given by the product of two smaller functions, say, \( u \) and \( v \), the product rule will help us differentiate it.
The rule states:\[(uv)' = u'v + uv'\]This means you take the derivative of the first function, \( u \), and multiply it by the second function, \( v \), as it is. Then, add the product of \( u \) and the derivative of \( v \).
In our specific exercise, \( y = \theta \sin(\log_{7} \theta) \) can be broken into:

• \( u = \theta \)
• \( v = \sin(\log_{7} \theta) \)

Applying the product rule involves finding \( u' \), which in this case is \( 1 \), and \( v' \), which requires the chain rule to solve.

Diving Into the Chain Rule

The chain rule is another fundamental technique in calculus, specifically used when you're differentiating a composite function.
This involves a function nested within another function, like \( \sin(\log_{7}\theta) \).For a composite function of the form \( f(g(x)) \), the derivative is given by the chain rule:\[\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\]Here's how it applies:

• Differentiate the outer function \( \sin(g) \), where the inner function \( g(x) = \log_{7} \theta \), gives \( \cos(g) \).


• Then, differentiate \( g(x) \) itself: using the property of logarithms, \( g'(x) = \frac{1}{\theta \ln 7} \).

Combine these results to get the derivative of \( v \), which contributes to the larger differentiation problem involving the product rule.

Exploring Logarithmic Differentiation

When confronted with functions involving logarithms, specifically a term like \( \log_{7}\theta \), simplifying the differentiation process often involves the change-of-base formula from logarithms.
Here's how it works: Convert \( \log_{7}\theta \) into natural logarithms for easier differentiation using the formula:
\[\log_{7}\theta = \frac{\ln \theta}{\ln 7}\]Now, differentiate the expression \( \log_{7}\theta \) with respect to \( \theta \) as it's vital for the chain rule:

• The derivative \( \frac{d}{d\theta}(\log_{7}\theta) = \frac{1}{\theta \ln 7} \).


• This result helps us find the necessary parts for the derivative of the inner function in the chain rule setting.

Logarithmic differentiation simplifies working with complex functions by emphasizing structural transformation, making differentiation clear and manageable.