derivative of log(x)

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29-03-2025

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The derivative of log(x) is a fundamental concept in calculus with wide-ranging applications. This article will explore how to derive this important result, providing a clear and comprehensive understanding for both beginners and those looking for a refresher. We'll cover different bases of logarithms and explore practical examples.

The Derivative of ln(x) (Natural Logarithm)

The simplest case involves the natural logarithm, denoted as ln(x), which has a base of e (Euler's number, approximately 2.718). Understanding this derivation is key to understanding other logarithmic derivatives.

The Limit Definition of the Derivative

The derivative of a function f(x) is defined as:

f'(x) = lim (h→0) [(f(x + h) - f(x)) / h]

Applying this to ln(x):

ln'(x) = lim (h→0) [(ln(x + h) - ln(x)) / h]

Using Logarithmic Properties

We can simplify this expression using the logarithmic property ln(a) - ln(b) = ln(a/b):

ln'(x) = lim (h→0) [ln((x + h) / x) / h]

Further simplification using the property a*ln(b) = ln(b^a):

ln'(x) = lim (h→0) [ln((1 + h/x)^(1/h))]

Now, let's manipulate the expression inside the limit. We can rewrite it as:

ln'(x) = lim (h→0) ln[(1 + h/x)^(x/h) * (1/x)]

Using properties of logarithms, we have:

ln'(x) = lim (h→0) [ln((1 + h/x)^(x/h)) + ln(1/x)]

As h approaches 0, (1 + h/x)^(x/h) approaches e. Therefore:

ln'(x) = ln(e) + ln(1/x) = 1 + ln(1/x)

However, this simplification didn't quite lead us to the correct answer. Let's go back and use a different approach.

Using the Chain Rule and Implicit Differentiation

Let y = ln(x). Then, by definition of the logarithm, we have:

e^y = x

Now we differentiate both sides with respect to x, using the chain rule on the left side:

e^y * (dy/dx) = 1

Solving for dy/dx (which is the derivative of ln(x)):

dy/dx = 1 / e^y

Since e^y = x, we substitute:

dy/dx = 1/x

Therefore, the derivative of ln(x) is 1/x.

The Derivative of log_b(x) (Logarithm with Base b)

For a logarithm with any base b, we can use the change of base formula:

log_b(x) = ln(x) / ln(b)

Since ln(b) is a constant, we can differentiate using the constant multiple rule:

d/dx [log_b(x)] = (1/ln(b)) * d/dx [ln(x)] = 1 / (x * ln(b))

Practical Examples

Example 1: Find the derivative of f(x) = 3ln(x) + x².

Using the sum rule and constant multiple rule: f'(x) = 3(1/x) + 2x = 3/x + 2x

Example 2: Find the derivative of g(x) = log₂(x).

Using the formula derived above: g'(x) = 1 / (x * ln(2))

Example 3: Application in Optimization

Suppose the cost function of producing x units is C(x) = 1000 + 50x + 10ln(x). The marginal cost is the derivative of the cost function:

C'(x) = 50 + 10/x

This tells us how much the cost increases with each additional unit produced.

Conclusion

The derivative of log(x) is a crucial concept in calculus. Whether dealing with the natural logarithm (ln(x)) or a logarithm with a different base (log_b(x)), understanding the derivation and its applications is essential for solving various mathematical problems, especially in areas like optimization and analysis. Remember the key results: d/dx[ln(x)] = 1/x and d/dx[log_b(x)] = 1/(x*ln(b)). This foundation will help you tackle more complex problems involving logarithmic functions.