Problem 57 - Calculus and its applications

Question

Differentiate. $$ y=-9 \ln x $$

Step-by-Step Solution

Answer

The derivative is \( -\frac{9}{x} \).

Step 1: Identify the Function Structure

The function given is \( y = -9 \ln x \). This is a logarithmic function multiplied by a constant, which is \( -9 \). The derivative of the natural logarithm function \( \ln x \) is known to be \( \frac{1}{x} \).

Step 2: Use the Constant Factor Rule

The constant factor rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. Thus, differentiate the function using this rule: \( \frac{d}{dx}(-9 \ln x) = -9 \frac{d}{dx}(\ln x) \).

Step 3: Differentiate \( \ln x \)

The derivative of \( \ln x \) is \( \frac{1}{x} \). Substitute this result into the expression from Step 2. Thus, \( \frac{d}{dx}(-9 \ln x) = -9 \times \frac{1}{x} \).

Step 4: Simplify the Expression

Multiply the constant \( -9 \) by \( \frac{1}{x} \). This simplifies to: \( \frac{d}{dx}(-9 \ln x) = -\frac{9}{x} \). This is the derivative of the given function.

Key Concepts

Logarithmic Functions

Logarithmic functions are special mathematical functions that can simplify solving for exponentials. The notation \( \ln(x) \) represents the natural log, which is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. Natural logarithms have several important properties that make them useful in calculus.

• Natural logs are the inverse of exponential functions, meaning \( e^{\ln(x)} = x \).
• The domain of \( \ln(x) \) is \( x > 0 \), since you can only take the logarithm of positive numbers.
• Logarithmic functions grow slowly, increasing without bound but at a decreasing rate.

Understanding logarithmic functions is crucial as they often appear in both pure and applied mathematics, especially when dealing with growth rates and compound interest.

Derivative of Natural Logarithm

The derivative of the natural logarithm function is an essential concept in calculus, especially when working with logarithmic differentiation. The derivative of \( \ln(x) \) is a straightforward formula:
\[ \frac{d}{dx} \ln(x) = \frac{1}{x} \].

Here's how it works:

• This formula implies that when you differentiate the natural log of \( x \), the result is \( \frac{1}{x} \).
• Think of it as the reciprocal of \( x \); as \( x \) grows, the derivative gets smaller.
• This behavior reflects the very gradual growth rate of the logarithmic function itself.

When differentiating more complex expressions involving \( \ln(x) \), simply remember this rule, and you'll be able to handle any necessary calculations with ease.

Constant Factor Rule

The constant factor rule is a handy shortcut for differentiating functions that involve a constant multiplied by a variable expression. It allows us to pull out the constant and focus on differentiating the variable part.

To apply the rule:

• Identify the constant multiplying the function.
• The derivative of the constant times a function is the constant times the derivative of the function.
• For example, in differentiating \( -9 \ln x \), the constant factor is \( -9 \).
• Thus, employing the constant factor rule: \[ \frac{d}{dx} (-9 \ln x) = -9 \cdot \frac{d}{dx} (\ln x) \].

This makes the differentiation process more manageable, especially with functions involving more complicated expressions, ensuring accurate calculations with minimal effort.