Problem 79 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

\( \frac{d}{d\theta}(y) = \sin(\log_7 \theta) + \frac{\cos(\log_7 \theta)}{\ln 7}. \)

Step 1: Identify the Inner and Outer Functions

The given function is composed of multiple functions: 1. The innermost function is the logarithm, \(\log_7 \theta\).2. The next function is the sine function, \(\sin(u)\), where \(u = \log_7 \theta\).3. The outermost product is \(\theta \cdot \sin(u)\).This defines a composite function that requires the product rule and the chain rule for differentiation.

Step 2: Apply the Product Rule

The product rule in differentiation is given by: \((uv)' = u'v + uv'\). Here, \(u = \theta\) and \(v = \sin(\log_7 \theta)\). First, compute the derivatives:- \(u' = \frac{d}{d\theta}(\theta) = 1\).- \(v' = \frac{d}{d\theta}(\sin(\log_7 \theta))\). Apply the product rule as follows:\(\frac{d}{d\theta}(y) = 1 \cdot \sin(\log_7 \theta) + \theta \cdot \left(\frac{d}{d\theta}(\sin(\log_7 \theta))\right)\).

Step 3: Apply the Chain Rule to Differentiate the Sine Function

To differentiate \(\sin(\log_7 \theta)\), use the chain rule: \(\frac{d}{dx}(\sin(u)) = \cos(u) \cdot \frac{du}{dx}\) where \(u = \log_7 \theta\). So, \(\frac{d}{d\theta}(\sin(\log_7 \theta)) = \cos(\log_7 \theta) \cdot \frac{d}{d\theta}(\log_7 \theta)\).

Step 4: Differentiate the Logarithmic Function

The derivative of the logarithmic function w.r.t. \(\theta\) is computed as follows: \(\log_7 \theta = \frac{\ln \theta}{\ln 7}\).Thus, \(\frac{d}{d\theta}(\log_7 \theta) = \frac{1}{\theta \ln 7}\).

Step 5: Substitute Back Into the Derivative

Substitute \(\frac{d}{d\theta}(\log_7 \theta) = \frac{1}{\theta \ln 7}\) obtained from Step 4 into the derivative from Step 3 to get:\(\frac{d}{d\theta}(\sin(\log_7 \theta)) = \cos(\log_7 \theta) \cdot \frac{1}{\theta \ln 7}\).

Step 6: Final Derivative Expression Using All Components

Substitute the result from Step 5 into the expression from Step 2:\(\frac{d}{d\theta}(y) = \sin(\log_7 \theta) + \theta \cdot \left(\cos(\log_7 \theta) \cdot \frac{1}{\theta \ln 7}\right)\).After simplifying, the expression reduces to:\(\frac{d}{d\theta}(y) = \sin(\log_7 \theta) + \frac{\cos(\log_7 \theta)}{\ln 7}.\)

Key Concepts

Product Rule

The product rule is a handy tool when you need to differentiate a product of two functions. If you have a function of the form \(y = u imes v\), where both \(u\) and \(v\) are functions of the same variable, the derivative of \(y\) is given by:

• \((uv)' = u'v + uv'\)

This rule is essential because it helps manage the differentiation of functions like \(\theta \cdot \sin(\log_{7}\theta)\), where you must deal with multiple operations.
Each term in the product, \(\theta\) and \(\sin(\log_{7}\theta)\), must be handled separately, and then the results can be put together using the product rule. It's like splitting tasks into manageable parts and solving each one, making complex problem-solving more straightforward.

Chain Rule

The chain rule is your go-to for differentiating composite functions, those nested functions where one function sits inside another. For a function \(y = f(g(x))\), the derivative is computed as:

• \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)

It's like peeling an onion: you take it layer by layer. In the provided exercise, the function \(\sin(\log_{7} \theta)\) is a perfect example.
First, you differentiate the outer layer, the sine function. Then, you multiply that result by the derivative of the inner layer, the logarithm function.
This way, the chain rule allows you to break down and simplify the differentiation process of multi-layered functions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, pop up quite frequently in calculus. Knowing how to differentiate these can make your life a lot easier. The fundamental rules are:

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

For the function \(\sin(\log_{7} \theta)\), when differentiating, you first handle the sine using its rule and the chain rule to deal with the logarithmic part inside.
It's essential to remember these basic derivatives, as they provide a foundation for more complex problems involving trigonometric expressions.

Logarithmic Differentiation

Logarithmic differentiation is a technique that can make differentiating complex expressions easier, especially when a variable is in both the base and the exponent. For logarithms, the derivative of \(\log_b(x)\) can be adjusted to the natural log as follows:

• \(\frac{d}{dx}(\log_b x) = \frac{1}{x \ln b}\)

This is particularly useful in our problem where we have \(\log_{7} \theta\). By converting the base 7 log into natural logs, you can efficiently differentiate using this rule, simplifying the overall differentiation process.
This method not only simplifies differentiating log functions but also integrates well with the chain rule when handling nested functions.