Problem 19 - Calculus and its applications

Question

Differentiate. $$ F(x)=\log (6 x-7) $$

Step-by-Step Solution

Answer

The derivative is \( F'(x) = \frac{6}{6x - 7} \).

Step 1: Identify the Function Type

The function given is a logarithmic function, specifically in the form of \( F(x) = \log(6x - 7) \). This means we will need to use the chain rule for differentiation because the inner function is linear.

Step 2: Recall the Derivative of Logarithmic Functions

The derivative of a logarithmic function \( \log(u) \) with respect to \( u \) is \( \frac{1}{u} \). We will also need to differentiate the inner function \( u = 6x - 7 \).

Step 3: Differentiate the Inner Function (Chain Rule)

First, differentiate the inner function \( u = 6x - 7 \), which gives \( \frac{du}{dx} = 6 \).

Step 4: Apply the Chain Rule

Using the chain rule, the derivative of \( F(x) = \log(6x - 7) \) is obtained by multiplying the derivative of the outside function by the derivative of the inside function: \[ F'(x) = \frac{1}{6x - 7} \cdot 6 \]

Step 5: Simplify the Derivative Expression

Simplify the expression obtained: \[ F'(x) = \frac{6}{6x - 7} \]

Key Concepts

Chain Rule in Differentiation

The chain rule is a powerful tool in calculus for finding the derivative of composite functions. Composite functions are those that have one function nested inside another. When dealing with such functions, it is crucial to break them down.
In the case of our function, \( F(x) = \log(6x - 7) \), the chain rule helps us differentiate by focusing on two distinct parts:

• The outer function, which is \( \log(u) \)
• The inner function, which is \( u = 6x - 7 \)

To apply the chain rule, you first differentiate the outer function and then multiply it by the derivative of the inner function.
This process is essential for maintaining accuracy when differentiating complex structures, allowing you to focus on smaller, more manageable pieces.

Understanding Logarithmic Functions

Logarithmic functions are widely used in various fields including science and engineering. They are the inverse of exponential functions and have their own unique properties. When differentiating them, the crucial aspect to remember is their fundamental derivative formula:
For a logarithmic function \( \log(u) \), its derivative with respect to \( u \) is \( \frac{1}{u} \).
This simple yet powerful rule is the cornerstone for logarithmic differentiation. In our exercise, recognizing this property was key in breaking down the differentiation process.
Thus, when faced with a logarithmic function, always identify its argument or inner part and apply this basic derivative rule.

Simplifying Derivative Expressions

After performing differentiation using the chain rule, one arrives at an expression that needs simplification. Simplifying derivatives makes them easier to interpret and utilize in practical applications.
In the case of the expression \( F'(x) = \frac{6}{6x - 7} \), simplification involves carefully following mathematical principles:

• Combine terms to create an understandable structure.
• Ensure that the expression is in its simplest form to avoid unnecessary complexity.

This final step is essential as it enhances clarity and makes further mathematical manipulation significantly more straightforward. Having a simplified expression also aids in communicating mathematical ideas effectively.