What Is The Derivative Of Ln5x

What is the Derivative of ln(5x)? A Comprehensive Guide

The derivative of ln(5x) is a fundamental concept in calculus, frequently encountered in various applications. Understanding how to derive this and similar logarithmic functions is crucial for mastering calculus and its applications in fields like physics, engineering, and economics. This comprehensive guide will walk you through the process step-by-step, explaining the underlying principles and offering additional examples to solidify your understanding.

Understanding the Chain Rule

Before we delve into finding the derivative of ln(5x), let's refresh our understanding of the chain rule. The chain rule is a crucial tool for differentiating composite functions – functions within functions. It states that the derivative of a composite function is the derivative of the outer function (with the inside function left alone) multiplied by the derivative of the inside function. Mathematically:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

This rule will be vital in solving our problem, as ln(5x) is a composite function: the natural logarithm function (ln) applied to the function 5x.

Deriving the Derivative of ln(5x)

Now, let's tackle the derivative of ln(5x). We'll apply the chain rule systematically.

1. Identify the Outer and Inner Functions:

   • Outer function: f(u) = ln(u) where u represents the inner function.
   • Inner function: g(x) = 5x

2. Find the Derivatives of the Outer and Inner Functions:

   • Derivative of the outer function: f'(u) = 1/u (The derivative of ln(u) is 1/u).
   • Derivative of the inner function: g'(x) = 5 (The derivative of 5x with respect to x is 5).

3. Apply the Chain Rule:

   Substitute the derivatives and the inner function into the chain rule formula:

   d/dx [ln(5x)] = f'(g(x)) * g'(x) = (1/(5x)) * 5

4. Simplify:

   The 5 in the numerator and denominator cancel out, leaving us with:

   d/dx [ln(5x)] = 1/x

Therefore, the derivative of ln(5x) is simply 1/x.

Understanding the Result: Implications and Applications

The result, 1/x, might seem surprisingly simple given the initial complexity. However, this simplicity highlights the power and elegance of calculus. Let's explore some implications and practical applications:

• Logarithmic Differentiation: This example showcases the elegance of logarithmic differentiation, a powerful technique used to simplify the differentiation of complex functions, especially those involving products, quotients, and powers. By taking the natural logarithm of a function before differentiating, often simplifies the process, as we've seen here.

• Marginal Analysis in Economics: In economics, the derivative represents the instantaneous rate of change. If ln(5x) represents a production function (where x is input and ln(5x) is output), then 1/x represents the marginal product – the additional output gained from an additional unit of input. This is incredibly useful for optimizing production and resource allocation.

• Growth and Decay Models: Natural logarithms are frequently used in modelling exponential growth and decay phenomena. The derivative plays a crucial role in analyzing the rate of growth or decay at any given point in time. For instance, if ln(5x) represents the population growth of a certain species, the derivative helps determine the rate of population increase.

• Solving Differential Equations: Differential equations are equations involving functions and their derivatives. Many differential equations involve logarithmic functions, and the knowledge of their derivatives is essential for solving these equations. These equations model numerous real-world phenomena, from the spread of diseases to the movement of objects under gravity.

Further Examples and Practice

To reinforce your understanding, let's consider a few more examples:

Example 1: Finding the derivative of ln(2x+1)

1. Outer function: f(u) = ln(u)
2. Inner function: g(x) = 2x + 1
3. Derivative of outer function: f'(u) = 1/u
4. Derivative of inner function: g'(x) = 2
5. Chain rule application: d/dx [ln(2x+1)] = (1/(2x+1)) * 2 = 2/(2x+1)

Example 2: Finding the derivative of ln(x²)

1. Outer function: f(u) = ln(u)
2. Inner function: g(x) = x²
3. Derivative of outer function: f'(u) = 1/u
4. Derivative of inner function: g'(x) = 2x
5. Chain rule application: d/dx [ln(x²)] = (1/x²) * 2x = 2/x

Example 3: A more complex example involving product rule: Find the derivative of x*ln(3x)

This example requires both the product rule and the chain rule.

1. Product Rule: d/dx[uv] = u dv/dx + v du/dx where u = x and v = ln(3x)
2. Derivative of u: du/dx = 1
3. Derivative of v: We need the chain rule here:
   • Outer function: f(u) = ln(u)
   • Inner function: g(x) = 3x
   • dv/dx = (1/(3x)) * 3 = 1/x
4. Applying the Product Rule: d/dx[x*ln(3x)] = x * (1/x) + ln(3x) * 1 = 1 + ln(3x)

These examples illustrate the versatility of the chain rule and how it extends to more complex scenarios involving logarithmic functions. Consistent practice with various examples will strengthen your understanding and ability to tackle more intricate problems.

Conclusion: Mastering Logarithmic Differentiation

Understanding the derivative of ln(5x), and more generally, logarithmic differentiation, is fundamental to mastering calculus. The simplicity of the result (1/x) shouldn't overshadow the importance of grasping the underlying principles – the chain rule and the properties of logarithmic functions. Through consistent practice and the exploration of diverse examples, you can develop a solid foundation in this crucial area of calculus, empowering you to tackle more advanced concepts and real-world applications with confidence. Remember to utilize the techniques discussed, particularly the chain rule, to solve a wide variety of logarithmic derivative problems. This consistent application will further solidify your understanding and build your problem-solving skills.