Question

What is \(\frac{d}{d x}(\ln x) ?\)

Short answer

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

Step-by-step solution

Understand the Function

We are given the natural logarithmic function \( \ln x \). Our task is to find its derivative with respect to \( x \).

Recall the Derivative Rule for \( \ln x \)

The derivative of \( \ln x \) with respect to \( x \) is a well-documented rule in calculus, which states: \[ \frac{d}{d x}(\ln x) = \frac{1}{x} \].

Evaluate the Derivative

Apply the derivative rule to the function \( \ln x \) to get: \[ \frac{d}{d x}(\ln x) = \frac{1}{x} \]. This result follows directly from the rule reiterated in Step 2.

Key concepts

Natural Logarithm

The natural logarithm, represented as \( \ln x \), is a fundamental mathematical function. It is the logarithm to the base \( e \), where \( e \) is an important mathematical constant approximately equal to 2.71828. The natural logarithm is used extensively in calculus, especially in problems involving exponential growth or decay.
Natural logarithms have unique properties that make them useful in various mathematical, scientific, and engineering contexts. One key property is that \( \ln(e) = 1 \) because \( e^1 = e \). Likewise, \( \ln(1) = 0 \) because \( e^0 = 1 \). This feature allows for simplifying expressions where exponential terms are present.
The natural logarithm is the inverse of the exponential function \( e^x \). For instance, if \( y = \ln(x) \), then \( e^y = x \). This relationship is crucial in solving equations where variables are exponents.

Calculus

Calculus is a branch of mathematics that studies change, focusing on derivatives and integrals. It provides tools for analyzing how things change and for calculating things that accumulate.
The derivative is the central concept within calculus that measures how a function changes at any given point. In simple terms, it describes the rate of change or the slope of the function at any point. For a function \( f(x) \), the derivative \( f'(x) \) or \( \frac{df}{dx} \), is a new function that tells us the slope of \( f(x) \) at every point.
In the context of the original exercise, finding \( \frac{d}{dx}(\ln x) \) involves applying the derivative rules of calculus to determine how the natural logarithm function changes with varying \( x \). Understanding how to compute derivatives is crucial in calculus, and it plays a significant role in fields such as physics, engineering, and economics, where rate of change is essential.

Derivative Rules

Derivative rules simplify the process of finding the derivative of functions. They provide a straightforward method to calculate how functions change, without needing to rely on the fundamental definition of a derivative each time.
For example, the derivative rule for the natural logarithm \( \ln x \) states that \( \frac{d}{dx}(\ln x) = \frac{1}{x} \). This rule is derived through the limit definition of a derivative, which can be more complex but results in this simple form.
Several other useful derivative rules include:

• The power rule: For \( x^n \), the derivative is \( nx^{n-1} \).
• The product rule: For functions \( u(x) \cdot v(x) \), the derivative is \( u'v + uv' \).
• The quotient rule: For functions \( \frac{u(x)}{v(x)} \), the derivative is \( \frac{u'v - uv'}{v^2} \).

Understanding these rules makes it easier to navigate complex calculus problems and ensures accuracy when determining how functions behave when their inputs change. Each rule applies to specific types of functions, and knowing when and how to use them is an important skill in calculus.