Derivative Of Ln

Unveiling the Derivative of the Natural Logarithm (ln x)

The natural logarithm, denoted as ln x, plays a crucial role in various fields, from calculus and physics to finance and computer science. Understanding its derivative is fundamental to solving numerous problems involving rates of change, optimization, and modeling. This article aims to provide a comprehensive exploration of the derivative of ln x, explaining its derivation, properties, and applications through detailed explanations and practical examples.

1. Defining the Natural Logarithm

Before diving into the derivative, let's establish a clear understanding of the natural logarithm. The natural logarithm, ln x, is the logarithm to the base e, where e is Euler's number, an irrational mathematical constant approximately equal to 2.71828. It is the inverse function of the exponential function, ex. This means that if y = ln x, then x = ey. This inverse relationship is key to understanding the derivative's derivation.

2. Deriving the Derivative using the First Principles

The derivative of a function f(x) is defined as the limit of the difference quotient as the change in x approaches zero:

f'(x) = limΔx→0 [(f(x + Δx) - f(x)) / Δx]

Applying this definition to f(x) = ln x:

1. Substitute: f'(x) = limΔx→0 [(ln(x + Δx) - ln(x)) / Δx]

2. Logarithm Property: Use the logarithm property ln(a) - ln(b) = ln(a/b):

f'(x) = limΔx→0 [ln((x + Δx)/x) / Δx]

3. Simplify:

f'(x) = limΔx→0 [ln(1 + Δx/x) / Δx]

4. Rewrite: Let u = Δx/x. Then, as Δx → 0, u → 0. Also, Δx = ux. Substituting:

f'(x) = limu→0 [ln(1 + u) / (ux)] = (1/x) limu→0 [ln(1 + u) / u]

5. Limit Evaluation: The limit limu→0 [ln(1 + u) / u] is a well-known limit that equals 1 (This can be proven using L'Hôpital's rule or the series expansion of ln(1+u)).

6. Final Result: Therefore, the derivative of ln x is:

f'(x) = 1/x

3. Understanding the Derivative's Significance

The derivative, 1/x, tells us the instantaneous rate of change of ln x at any point x. For example, at x = 1, the slope of the tangent to the curve y = ln x is 1. At x = 2, the slope is 1/2, and so on. The derivative is always positive for x > 0, indicating that the natural logarithm is a strictly increasing function in its domain (x > 0).

4. Practical Applications

The derivative of ln x is crucial in various applications:

Optimization Problems: Finding maxima or minima of functions involving natural logarithms.
Related Rates: Determining the rate of change of one variable with respect to another when both are related through a logarithmic function.
Economics and Finance: Calculating marginal utilities, growth rates, and other economic indicators.
Physics: Solving problems involving exponential decay or growth.

Example: Suppose the growth of a bacterial population is modeled by the equation N(t) = 1000 ln(t + 1), where N(t) is the population at time t (in hours). The rate of growth at t = 2 hours is given by the derivative N'(t) = 1000/(t+1). Therefore, at t = 2, the growth rate is N'(2) = 1000/3 ≈ 333.33 bacteria per hour.

5. Chain Rule with ln x

When dealing with composite functions involving the natural logarithm, the chain rule is essential. If y = ln(u(x)), then the derivative is given by:

dy/dx = (1/u(x)) (du/dx)

Example: If y = ln(x² + 1), then u(x) = x² + 1, and du/dx = 2x. Thus, dy/dx = (1/(x² + 1)) 2x = 2x/(x² + 1).

Conclusion

The derivative of ln x, being simply 1/x, provides a powerful tool for analyzing and solving a wide array of problems involving logarithmic functions. Its simplicity belies its importance in various scientific and mathematical disciplines. Understanding its derivation and application using the chain rule allows for proficient problem-solving across numerous fields.

FAQs:

1. What is the domain of ln x? The domain of ln x is (0, ∞), meaning x must be greater than zero.

2. What is the derivative of ln(ax)? Using the chain rule, the derivative is a/x.

3. Can I use L'Hôpital's rule to derive the derivative of ln x? Yes, L'Hôpital's rule can be used to evaluate the limit in step 5 of the derivation.

4. What is the integral of 1/x? The integral of 1/x is ln|x| + C, where C is the constant of integration. The absolute value is necessary to account for the domain of ln x.

5. How does the derivative of ln x relate to the derivative of ex? They are inverse functions, and their derivatives reflect this relationship. The derivative of ex is ex, highlighting the unique property of the exponential function.

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