Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ \frac{d y}{d x} \text { if } y=3 \ln x $$

Step-by-Step Solution

Answer

The derivative is \( \frac{3}{x} \).

Step 1: Identify the function and rule

The function given is \( y = 3 \ln x \). This is a logarithmic function, and to find the derivative, we use the chain rule as well as the derivative of the natural logarithm function \( \ln x \).

Step 2: Apply the derivative rule for natural logarithm

The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \). Since \( y = 3 \ln x \), we apply the constant multiple rule which states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

Step 3: Compute the derivative

Take the constant 3 out of the differentiation, so: \( \frac{d}{dx} [3 \ln x] = 3 \times \frac{d}{dx} [\ln x] \). Continuing with the derivative, \( \frac{d}{dx} [\ln x] = \frac{1}{x} \), so the derivative of the entire function is \( 3 \times \frac{1}{x} = \frac{3}{x} \).

Key Concepts

Chain Rule

The chain rule is an essential calculus method used to find the derivative of composite functions, which is a function made up of two or more nested functions. Imagine you have a function within another function, like a set of nested dolls. The outer function wraps around the inner one. The chain rule helps you differentiate the outer function while accounting for the changes in the inner one.
For example, if you have a function \( f(g(x)) \), where \( f \) is the outer function and \( g \) is the inner one, then the chain rule states:

• The derivative \( f'(g(x)) \cdot g'(x) \) must be calculated.

Essentially, this allows you to "chain" together the derivatives of the functions involved. Understanding the chain rule is crucial because it simplifies differentiation, especially when dealing with more complex expressions.
In the exercise given, the chain rule wasn't directly applied since \( 3 \ln x \) doesn't form a composite function needing it. However, recognizing when to use the chain rule can help solve another frequent differentiation challenges you might encounter.

Logarithmic Functions

Logarithmic functions, especially the natural logarithm function \( \ln x \), appear frequently in calculus. The natural logarithm is the inverse of the exponential function \( e^x \). It is preferred in many mathematical applications because of its unique properties, such as simplifying complex equations involving exponents.
In differentiation, the natural logarithm function's derivative has a straightforward property:

• The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \).

This makes it simple to compute compared to other logarithmic bases. For instance, in the exercise \( y = 3 \ln x \), the derivative is computed by first identifying \( \ln x \) and then applying the constant multiple rule. Understanding how logarithmic functions work and their derivatives helps streamline these calculations and provides a clearer path when you deal with more layered functions involving logarithms.

Derivatives

Derivatives are one of the cornerstones of calculus. They describe the rate at which a function changes at any given point, effectively representing the function's slope. Understanding derivatives involves several rules and methods to differentiate various types of functions, such as polynomial, exponential, and logarithmic functions.
For the given function \( y = 3 \ln x \), the derivative helps to understand how \( y \) changes as \( x \) changes. Using fundamental differentiation rules:

• We apply the derivative of \( \ln x \) which is \( \frac{1}{x} \),
• Then multiply by the constant \( 3 \) from the original function.

This results in the expression \( \frac{3}{x} \) as the derivative. Each differentiation rule, whether for constant, power, or more complex combinations, assists in solving real-world problems related to motion, growth, and changes over time. Mastering derivatives significantly boosts your analytical capabilities in understanding and interpreting the behavior of mathematical functions graphically and numerically.