Question

For the following exercises, find the derivative \(\frac{d y}{d x}\). $$ y=\ln (2 x) $$

Short answer

The derivative is \( \frac{1}{x} \).

Step-by-step solution

Identify the Type of Function

The function given is a natural logarithm function, \( y = \ln(2x) \). Identifying the function helps us know what rules to apply in differentiation.

Recall the Derivative Rule for Logarithms

The derivative of a natural logarithm, \( \ln(u) \), with respect to \( x \) is \( \frac{1}{u} \frac{du}{dx} \). This is what we will apply to the function \( \ln(2x) \).

Differentiate the Function Using the Chain Rule

The inner function \( u \) is \( 2x \), and its derivative \( \frac{du}{dx} \) is 2. Applying the chain rule, the derivative of \( \ln(2x) \) is \( \frac{1}{2x} \times 2 = \frac{2}{2x} \).

Simplify the Result

Simplifying \( \frac{2}{2x} \) gives \( \frac{1}{x} \). This is the derivative \( \frac{dy}{dx} \).

Key concepts

Natural Logarithm

The natural logarithm is a special type of logarithm which has some unique properties. Unlike the common logarithm, which uses a base of 10, the natural logarithm uses the mathematical constant \( e \) (approximately 2.718) as its base. This function is represented as \( \ln(x) \). The natural logarithm has fascinating uses in both pure and applied mathematics, especially in problems relating to exponential growth like those in finance and physics.

In this context, when you see a function like \( y = \ln(2x) \), it signifies a transformation that involves the scaling of \( x \) by 2 before applying the natural logarithm. This scaling impacts the differentiation process because it requires using the chain rule, which involves differentiating the outer function and then multiplying by the derivative of the inner function.

Chain Rule in Calculus

The chain rule is a fundamental technique in calculus used to differentiate composite functions. A composite function is essentially a function inside another function, denoted generally as \( f(g(x)) \). The chain rule provides a method to differentiate these by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

When dealing with the function \( y = \ln(2x) \), \( \ln \) is the outer function and \( 2x \) is the inner function. The chain rule states:

• First, take the derivative of the outer function \( \ln(u) \) using \( \frac{1}{u} \).
• Next, multiply this by the derivative of the inner function. Here, \( u = 2x \) and \( \frac{du}{dx} = 2 \).

This gives \( \frac{1}{2x} \times 2 = \frac{2}{2x} \). Simplifying this expression results in \( \frac{1}{x} \), which is the derivative of the original function. The chain rule eases the process of dealing with nested functions, allowing us to solve them step by step.

Function Differentiation

Function differentiation refers to the process of determining the rate at which a function is changing at any given point. This is expressed as the derivative of the function. Derivatives are a core concept in calculus, and they model how a change in one variable influences another.

When differentiating an expression like \( y = \ln(2x) \), you must not only recognize it as a logarithmic function but also apply rules appropriate to it for differentiation. The steps generally involve identifying the function type, applying rules relevant to that function, and simplifying the expression.

For \( y = \ln(2x) \), we see it as a natural logarithm, which needs the logarithmic differentiation rule \( \frac{1}{u} \frac{du}{dx} \). This makes differentiation a process of breaking down functions into simpler parts, calculating the derivative of each, and combining the results to find how the original function changes in relation to its variable.