Problem 61 - Calculus and its applications

Question

Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given. $$ y=\log _{3}(\log x) $$

Step-by-Step Solution

Answer

The derivative of the function is \( y' = \frac{1}{x \ln 3 \ln 10 \cdot \log x} \).

Step 1: Understand the function

We need to differentiate the function \( y = \log_3(\log x) \). This is a composite function involving a logarithm with base 3 and another logarithm with natural base.

Step 2: Apply Change of Base Formula

The usual derivative rules are more easily applied to natural logarithms, so convert the base 3 logarithm using the change of base formula: \( \log_3(u) = \frac{\ln u}{\ln 3} \), where \( u = \log x \). The function becomes \( y = \frac{\ln(\log x)}{\ln 3} \).

Step 3: Differentiate using the Constant Factor Rule

Notice that \( \frac{1}{\ln 3} \) is a constant factor, so differentiate \( y = \frac{1}{\ln 3} \cdot \ln(\log x) \) by finding the derivative of \( \ln(\log x) \).

Step 4: Differentiate \( \ln(\log x) \) using the Chain Rule

Let \( u = \log x \). Then \( y = \ln(u) \) and \( \frac{dy}{du} = \frac{1}{u} \). Now differentiate \( u = \log x \): \( \frac{du}{dx} = \frac{1}{x \ln 10} \) since \( \log x = \frac{\ln x}{\ln 10} \).

Step 5: Apply Chain Rule

Using the chain rule, the derivative is \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{\log x} \cdot \frac{1}{x \ln 10} = \frac{1}{x \ln 10 \cdot \log x} \).

Step 6: Combine the results and simplify

Combine the constant factor from Step 3: \( \frac{y'}{1} = \frac{1}{\ln 3} \cdot \frac{1}{x \ln 10 \cdot \log x} \). Thus, the derivative is \( y' = \frac{1}{x \ln 3 \ln 10 \cdot \log x} \).

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used mostly when a function is not explicitly defined. For example, if you have an equation involving both \(x\) and \(y\), and it’s not easy or possible to solve for \(y\) explicitly, implicit differentiation allows us to differentiate both sides of the equation with respect to \(x\) and find \(\frac{dy}{dx}\) directly.

Here's the gist of how it works:

• Differentiate both sides of the equation with respect to \(x\).
• When differentiating terms involving \(y\), pretend \(y\) is a function of \(x\), using the chain rule and multiplying by \(\frac{dy}{dx}\).
• By isolating \(\frac{dy}{dx}\), you solve for the derivative.

In our example, although the differentiation was explicit due to isolated \(y\), understanding implicit differentiation is crucial when dealing with more complex equations where \(y\) explicitly depends on \(x\).

Composite Function

A composite function is a function made by combining two or more functions. For instance, if \(f(x)\) and \(g(x)\) are two functions, the composite \(f(g(x))\) is formed by applying \(g\) first and then \(f\). This is a key concept as most real-world problems involve layers of operations, much like peeling an onion.

In the expression \(y = \log_3(\log x)\), \(\log x\) is nested inside \(\log_3(\cdot)\), making the entire function composite:

• First, you find the logarithm of \(x\).
• Then, you apply the base 3 logarithm on the result.

Handling composite functions during differentiation often requires the chain rule, which allows you to differentiate layer by layer, unraveling the composition slowly.

Thus, understanding the nature of composite functions is essential for differentiation and dealing with complex nested functions in calculus.

Change of Base Formula

The change of base formula is vital when dealing with logarithms of different bases. It allows us to convert a logarithm of one base into another base, typically into a natural logarithm for simpler calculation.

For any logarithm \(\log_b(x)\), the formula is:\[\log_b(x) = \frac{\ln(x)}{\ln(b)}\]This formula plays a significant role because derivative rules of logarithmic functions are more commonly known in terms of natural logarithms \(\ln\). Therefore, converting any logarithm into a form involving \(\ln\) simplifies differentiation.

In our example, converting \(\log_3(\log x)\) into \(\frac{\ln(\log x)}{\ln 3}\) using the change of base formula makes the derivative computation a lot easier. This is because now we can directly apply derivative rules for natural logarithms.

This formula acts like a translator, making mathematical expressions more comfortable to work with, especially when entering the world of calculus and differentiation.