Problem 23 - Calculus and its applications

Question

Differentiate. $$ f(x)=4 \log _{7}(\sqrt{x}-2) $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = 4 \log_7(\sqrt{x} - 2) \) is \( \frac{2}{\sqrt{x} \ln(7)(\sqrt{x} - 2)} \).

Step 1: Recognize the Function Composition

The function given is \( f(x) = 4 \log_7(\sqrt{x} - 2) \), which is a composition of functions: a constant scaling (\(4\)), a logarithm with a base of 7, and an inside function \( u(x) = \sqrt{x} - 2 \).

Step 2: Differentiate the Outer Function

Since the logarithm is with respect to base 7, use the differentiation rule, \( \frac{d}{dx}[\log_b(u)] = \frac{1}{u \ln(b)} \frac{du}{dx} \). The derivative of the outer function \( \log_7 u(x) \) is \( \frac{1}{u(x) \ln(7)} \).

Step 3: Differentiate the Inner Function

The inner function is \( u(x) = \sqrt{x} - 2 \). Differentiate \( \sqrt{x} - 2 \) to get \( \frac{d}{dx}[u(x)] = \frac{1}{2\sqrt{x}} \) because \( \frac{d}{dx}[\sqrt{x}] = \frac{1}{2\sqrt{x}} \) and \( -2 \) is a constant whose derivative is 0.

Step 4: Apply the Chain Rule

Apply the chain rule, which involves multiplying the derivative of the outer function by the derivative of the inner function. Thus, the derivative is \( \frac{4}{(\sqrt{x} - 2) \ln(7)} \cdot \frac{1}{2\sqrt{x}} \).

Step 5: Simplify the Expression

Combine the expressions from the chain rule application: \( \frac{4}{(\sqrt{x} - 2) \ln(7)} \cdot \frac{1}{2\sqrt{x}} = \frac{2}{\sqrt{x} \ln(7)(\sqrt{x} - 2)} \). This is the derivative of the function \( f(x) \).

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus, particularly when dealing with composite functions. It allows us to differentiate the composition of two or more functions. When you see a problem where one function is inside another, like in our example where the logarithm function wraps the expression \( \sqrt{x} - 2 \), the chain rule becomes highly relevant.

Here, the chain rule tells us to

• First, differentiate the outer function \( \log_7(u) \) with respect to its argument \( u \).
• Then multiply by the derivative of the inner function \( u(x) = \sqrt{x} - 2 \).

Why is it useful?
The chain rule simplifies the differentiation process when dealing with nested functions, ensuring accuracy in complex problems. It preserves the integrity of the derivative by logically breaking down the differentiation into manageable steps.

Steps in this exercise:

• The outer function is \( \log_7(\cdot) \) and we find its derivative as \( \frac{1}{u \ln(7)} \).
• The inner function is \( \sqrt{x} - 2 \) and its derivative is \( \frac{1}{2\sqrt{x}} \).
• Using the chain rule, multiply these derivatives to find the overall derivative.

Logarithmic Differentiation

Logarithmic differentiation is particularly useful when dealing with complex functions involving logarithms, as it simplifies the finding of derivatives. In our exercise, we encountered a logarithm with a non-standard base — base 7 — which means we must include the natural logarithm, \( \ln(7) \), when differentiating.

Formula for Differentiating Logarithms: The rule is:

• If \( y = \log_b(u) \), then \( \frac{dy}{dx} = \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \).
• This serves to convert the derivative into a form that utilizes natural logarithms, \( \ln(b) \), easing the differentiation process.

How was it applied?
In our example, since \( f(x) = 4 \log_7(\sqrt{x} - 2) \), identifying the base other than \( e \) required using \( \ln(7) \) to find the derivative of the outer function.
Why choose logarithmic differentiation?

• It's effective for handling complex logarithmic expressions.
• It's particularly handy when base changes are involved, allowing us to incorporate natural logarithms into the calculation.

Composite Functions

Composite functions are functions within functions. This means that one function provides an input to another. In the exercise given, you're dealing with a logarithmic function where its argument is itself a function \( \sqrt{x} - 2 \).

Understanding Composite Structure:

• Identifying such structures is critical. In our function, \( f(x) = 4 \log_7(\sqrt{x} - 2) \), the composite nature is explicit since \( \log_7(\cdot) \) needs \( \sqrt{x} - 2 \) to process its input.
• Recalling that composite functions necessitate the use of the chain rule for differentiation will help you tackle similar calculus problems with ease.

Why Composite Functions Matter in Differentiation:

• They indicate layers of operations, each contributing to the overall derivative.
• Recognizing that a function is composite alerts you to break down differentiation into manageable sub-problems.
• Composite functions are everywhere in advanced mathematics, making familiarity with their structure indispensable for calculus success.

Therefore, a firm grasp of recognizing and differentiating these layered functions is key to mastering calculus differentiation techniques.