What Is The Derivative Of Ln 5x

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

The natural logarithm, denoted as ln(x), is a fundamental concept in calculus and has numerous applications in various fields. Understanding its derivative is crucial for solving many mathematical problems. This comprehensive guide will explore the derivative of ln(5x), providing a step-by-step explanation, practical examples, and related concepts.

Understanding the Chain Rule

Before diving into the derivative of ln(5x), let's review the chain rule, a fundamental rule in differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function (with the inside function left alone) times the derivative of the inside function. Mathematically, if we have a function y = f(g(x)), then its derivative is given by:

dy/dx = f'(g(x)) * g'(x)

This rule is essential for finding the derivative of ln(5x) because ln(5x) is a composite function where the outer function is ln(u) and the inner function is u = 5x.

Deriving the Derivative of ln(5x)

Now, let's find the derivative of ln(5x) using the chain rule.

1. Identify the outer and inner functions:

   • Outer function: f(u) = ln(u)
   • Inner function: g(x) = 5x

2. Find the derivatives of the outer and inner functions:

   • Derivative of the outer function: f'(u) = 1/u (Recall that the derivative of ln(x) is 1/x)
   • Derivative of the inner function: g'(x) = 5

3. Apply the chain rule:

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

4. Simplify:

   • d/dx [ln(5x)] = 5 / (5x) = 1/x

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

Illustrative Examples

Let's solidify our understanding with some examples:

Example 1: Find the slope of the tangent line to the curve y = ln(5x) at x = 1.

To find the slope, we need to evaluate the derivative at x = 1. Since the derivative is 1/x, the slope at x = 1 is 1/1 = 1.

Example 2: Find the derivative of f(x) = x²ln(5x).

This requires applying the product rule along with the chain rule. The product rule states that the derivative of a product of two functions, u(x)v(x), is u'(x)v(x) + u(x)v'(x).

• u(x) = x², u'(x) = 2x
• v(x) = ln(5x), v'(x) = 1/x (as derived above)

Applying the product rule:

f'(x) = (2x)(ln(5x)) + (x²)(1/x) = 2x ln(5x) + x

Example 3: A more complex scenario

Let's consider a function where ln(5x) is nested within another function. For instance, find the derivative of h(x) = e^(ln(5x)).

We can simplify this first: e^(ln(5x)) = 5x (because e and ln are inverse functions). Therefore, the derivative of h(x) = 5x is simply 5.

Alternatively, using the chain rule directly:

• Outer function: f(u) = e^u

• Inner function: g(x) = ln(5x)

• f'(u) = e^u

• g'(x) = 1/x

h'(x) = e^(ln(5x)) * (1/x) = 5x * (1/x) = 5

Practical Applications

The derivative of logarithmic functions, including ln(5x), finds extensive use in various fields:

• Economics: Used in analyzing growth rates, elasticity of demand, and optimization problems.
• Physics: Applied in solving differential equations related to radioactive decay, heat transfer, and other physical phenomena.
• Engineering: Used in modeling and analysis of systems involving exponential growth or decay.
• Computer Science: Used in algorithm analysis, particularly in analyzing the time complexity of algorithms.
• Finance: Used extensively in compound interest calculations and financial modeling.

Understanding the Natural Logarithm

The natural logarithm, ln(x), is the logarithm to the base e, where e is Euler's number, approximately equal to 2.71828. It's the inverse function of the exponential function e^x. This means that if y = ln(x), then x = e^y.

Key properties of the natural logarithm include:

• ln(1) = 0
• ln(e) = 1
• ln(xy) = ln(x) + ln(y)
• ln(x/y) = ln(x) - ln(y)
• ln(x^r) = r ln(x)

Related Differentiation Rules

Besides the chain rule and product rule, other rules of differentiation are often used in conjunction with finding the derivative of logarithmic functions:

• Quotient Rule: Used when differentiating a function that is a quotient of two functions.
• Power Rule: Used when differentiating functions of the form x^n.

Advanced Topics

For a deeper understanding, one can explore more advanced concepts like:

• Implicit Differentiation: Used to find the derivative of functions where the dependent variable is not explicitly expressed as a function of the independent variable.
• Logarithmic Differentiation: A technique useful for differentiating functions involving products, quotients, and powers of functions.
• Partial Derivatives: Used when dealing with functions of multiple variables.

Conclusion

Finding the derivative of ln(5x) using the chain rule is straightforward. Understanding this fundamental concept, along with the other differentiation rules, is crucial for solving numerous mathematical problems across various disciplines. This comprehensive guide has provided a detailed explanation, illustrative examples, and explored the broader context of logarithmic functions and their applications. Remember to practice regularly to strengthen your understanding and problem-solving abilities. By mastering these concepts, you'll be well-equipped to tackle more complex problems in calculus and related fields.