When solving calculus problems, you might encounter the natural logarithm function, ln. The derivative of ln 5x is an essential concept that you should know. In this article, we will cover the steps to find the derivative of ln 5x and its applications.

## What is the Natural Logarithm Function?

The natural logarithm function, ln, is the inverse of the exponential function, e. It is a logarithmic function with base e, where e is approximately 2.71828. The notation of the natural logarithm function is ln x, where x is the argument of the function. The natural logarithm function has many applications in mathematics, science, and engineering.

## What is the Derivative?

The derivative is a crucial concept in calculus that describes the rate of change of a function with respect to its input variable. It is the instantaneous rate of change of a function at a particular point. The derivative of a function f(x) is denoted by f'(x) or dy/dx.

## Derivative of ln x

The derivative of ln x is 1/x. To find the derivative of ln x, we can use the logarithmic differentiation technique. The logarithmic differentiation technique is a method to find the derivative of a function that is a product, quotient, or power of other functions.

Let f(x) = ln x. Taking the natural logarithm of both sides, we get:

ln[f(x)] = ln[ln x]

Using the chain rule, we get:

f'(x)/f(x) = 1/ln x

Multiplying both sides by f(x), we get:

f'(x) = (1/x)ln x

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

## Derivative of ln 5x

The derivative of ln 5x is (1/x). To find the derivative of ln 5x, we can use the chain rule. The chain rule is a method to find the derivative of a composite function.

Let f(x) = ln 5x. Using the chain rule, we get:

f'(x) = (1/(5x)) * 5

f'(x) = (1/x)

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

## Applications of Derivative of ln 5x

The derivative of ln 5x has many applications in mathematics, science, and engineering. One of the applications is in exponential growth and decay problems. Exponential growth and decay problems involve a quantity that changes over time, and the rate of change is proportional to the quantity itself.

For example, suppose the population of a city is growing at a rate proportional to the population itself. The population of the city after t years can be represented by the equation:

P(t) = P(0)e^(rt)

Where P(0) is the initial population, r is the growth rate, and e is the base of the natural logarithm function. Using the derivative of ln 5x, we can find the growth rate r by taking the derivative of both sides of the equation:

P'(t)/P(t) = r

Using the chain rule and the derivative of ln 5x, we get:

P'(t)/P(t) = (1/P(t))dP(t)/dt = (1/P(0)e^(rt))(d/dt)(P(0)e^(rt)) = (r)

Therefore, the growth rate r is equal to the derivative of ln 5x.

## Conclusion

The derivative of ln 5x is (1/x). It can be found using the chain rule or logarithmic differentiation technique. The derivative of ln 5x has many applications in mathematics, science, and engineering, such as exponential growth and decay problems. Understanding the concept of the derivative of ln 5x is essential for solving calculus problems.