The Derivative of Natural Logarithm ln x and Its Implications

The Derivative of Natural Logarithm ln x and Its Implications

Understanding the derivative of the natural logarithm function, ln x, is a fundamental concept in calculus. This article will explore why the derivative of (ln{x}) is equal to (frac{1}{x}). We will delve into the proof using the limit definition of a derivative and show how this result is consistent with the Fundamental Theorem of Calculus.

Why the Derivative of Natural Logarithm is Equal to 1/x

The derivative of the natural logarithm function, (ln{x}), being equal to (frac{1}{x}) for all positive values of (x), is a fundamental principle in calculus. This can be proven using the limit definition of a derivative.

Using the Limit Definition of a Derivative

Step 1: Define the Function

Consider the function (f(x) ln{x}). The derivative of this function, evaluated at any point (x), can be defined using the limit:

[f'(x) lim_{h to 0} frac{ln(x h) - ln(x)}{h}]

Step 2: Simplify Using Logarithm Properties

To simplify this expression, we use the logarithm property that (ln(a) - ln(b) ln(frac{a}{b})):

[f'(x) lim_{h to 0} frac{ln(frac{x h}{x})}{h}]

[f'(x) lim_{h to 0} frac{ln(1 frac{h}{x})}{h}]

Step 3: Adjust the Limit Expression

To simplify the expression further, we let (k frac{h}{x}), so as (h to 0), (k to 0). The expression becomes:

[f'(x) lim_{k to 0} frac{ln(1 k)}{kx}]

Step 4: Recognize a Known Limit

The limit (lim_{k to 0} frac{ln(1 k)}{k}) is a well-known limit and is equal to 1. Thus, we have:

[f'(x) frac{1}{x} cdot lim_{k to 0} frac{ln(1 k)}{k}]

[f'(x) frac{1}{x} cdot 1]

[f'(x) frac{1}{x}]

Implications and Applications

The result that the derivative of (ln{x}) is (frac{1}{x}) has significant implications in both calculus and economics. In economics, it is used to understand the concept of marginal cost and marginal revenue.

The Fundamental Theorem of Calculus

By the Fundamental Theorem of Calculus, if we know that the derivative of (ln{x}) is (frac{1}{x}), then the antiderivative of (frac{1}{x}) must be (ln{x}). This can be formally written as:

[int frac{1}{x} , dx ln{|x|} C]

Understanding Derivatives and Marginal Concept

Many people wonder why the concept of derivative or marginal in economics is related to the limiting process. This is because the derivative represents the rate of change of a function, which is essentially a scaling factor based on the infinitesimal change in the input.

Conclusion

In conclusion, the derivative of the natural logarithm function (ln{x}) being equal to (frac{1}{x}) is a result of its mathematical properties and the fundamental principles of calculus. This result is not only crucial for understanding the antiderivative of (frac{1}{x}) but also provides insights into the limiting process and its applications in various fields, including economics.