Problem 28 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \(y=\log _{3} r \cdot \log _{9} r\)

Step-by-Step Solution

Answer

The derivative is \(\frac{\ln r}{r (\ln 3)^2}\).

Step 1: Rewrite Using Logarithm Properties

Express the logarithm to different bases using the change of base formula: \[\log_{3}r = \frac{\ln r}{\ln 3} \quad \text{and} \quad \log_{9}r = \frac{\ln r}{\ln 9}.\]Thus, the function becomes:\[y = \left(\frac{\ln r}{\ln 3}\right) \cdot \left(\frac{\ln r}{\ln 9}\right) = \frac{(\ln r)^2}{\ln 3 \cdot \ln 9}.\]

Step 2: Simplify the Expression

Simplify the expression by recognizing that \(\ln 9 = \ln(3^2) = 2\ln 3\). Substitute this into the function:\[y = \frac{(\ln r)^2}{\ln 3 \cdot 2 \ln 3} = \frac{(\ln r)^2}{2 (\ln 3)^2}.\]

Step 3: Differentiate Using Chain Rule

Determine the derivative of \(y = \frac{1}{2 (\ln 3)^2} (\ln r)^2\) with respect to \(r\). Using the chain rule, we have:\[\frac{dy}{dr} = \frac{1}{2 (\ln 3)^2} \cdot 2 \ln r \cdot \frac{1}{r} = \frac{\ln r}{r (\ln 3)^2}.\]

Step 4: Simplify the Result

Simplify the expression obtained from differentiation:\[\frac{dy}{dr} = \frac{\ln r}{r (\ln 3)^2}.\] This is the simplified form of the derivative of \(y\) with respect to \(r\).

Key Concepts

Logarithm Properties

Understanding logarithm properties is crucial for solving problems involving logarithms. Essentially, these are rules or theorems that simplify the computation and manipulation of logarithms. Key properties include:

• Product Rule: The logarithm of a product is the sum of the logarithms, \[\log_b(xy) = \log_bx + \log_by.\]


• Quotient Rule: The logarithm of a quotient is the difference of the logarithms, \[\log_b\left(\frac{x}{y}\right) = \log_bx - \log_by.\]


• Power Rule: The logarithm of a power is the exponent times the logarithm of the base, \[\log_b(x^k) = k\log_bx.\]

These properties help in transforming complex logarithmic expressions into simpler forms, making differentiation more manageable.

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. A composite function is one that is made up of two or more functions, where the output of one function becomes the input to another. The chain rule states that the derivative of a composite function \( f(g(x)) \) is the derivative of \( f \) with respect to \( g(x) \) times the derivative of \( g(x) \) with respect to \( x \):

• \[\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x).\]

For instance, in our exercise, \( \ln r \) is a composite function when squared. The chain rule was applied to differentiate \((\ln r)^2\), which is a product of \(\ln r\) with itself.

Change of Base Formula

The change of base formula is a handy technique to convert logarithms from one base to another. This is particularly useful when differentiating functions with logs of uncommon bases. The change of base formula is given by:

• \[\log_b a = \frac{\ln a}{\ln b}.\]

In this formula, \(\ln\) represents the natural logarithm, which is a standard logarithm with base \(e\), the mathematical constant \(e \approx 2.718\). By converting the logarithms \(\log_3 r\) and \(\log_9 r\) into natural logarithms, the derivative becomes easier to find.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any given point and is a fundamental concept in calculus. For the expression \(y = \log_3 r \cdot \log_9 r\), different calculus rules were employed:

• Power Rule: \(f(x) = x^n \Rightarrow f'(x) = nx^{n-1}.\)


• Chain Rule: As detailed before, it helps in handling complex products or compositions.


• Logarithmic Differentiation: Useful when logs are involved, simplifying the process substantially.

With these tools, we calculated the derivative \(\frac{dy}{dr}\) effectively.

Simplifying Expressions

Simplifying expressions is a key skill in mathematics that makes computation more straightforward and results easier to interpret. In the given exercise, the expression \(y = \frac{(\ln r)^2}{2 (\ln 3)^2}\) was simplified using several steps:

• Recognizing that \(\ln 9 = \ln (3^2) = 2 \ln 3\).


• Converting the product \((\ln r)^2\) into a simplified form divided by \(2(\ln 3)^2\).

This simplification process helped in obtaining a clearer and cleaner derivative form, \(\frac{\ln r}{r (\ln 3)^2}\), assisting both in calculations and understanding the problem at hand.