# Derivative of ln5x

Find about the derivative of ln 5x along with how to verify the derivative of ln5x by using implicit differentiation and product rule.

Alan Walker-

Published on 2023-05-26

## Introduction to the derivative of ln5x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln5 x can be calculated by following the rules of differentiation. Or, we can directly find the derivative of ln(x) by applying the first principle of differentiation. In this article, you will learn what the ln 5x derivative is and how to calculate the ln(5x) derivative by using different approaches.

## What is the derivative of ln 5x?

The derivative of ln5x with respect to x is equal to 1/x.  This can be expressed as d/dx ln(5x). It represents the rate of change of the natural logarithmic function ln(x) and is written as:

$\ln 5x=\log_e 5x$

The expression loge(5x) represents the logarithm of 5x with base e. This is a useful way to express the natural logarithm of 5x, because it helps us to easily calculate its derivative and understand its relationship with other mathematical functions.

## Derivative ln(5x) formula

The differentiation of ln(5x) is equal to 1/x. It can be calculated same as the derivative of ln(x). Applying this formula to ln(5x), you can simplify it to:

$\frac{d}{dx}(\ln 5x) =\frac{1}{x}$

This means that the derivative of ln(5x) is equal to the reciprocal of x. It means that the rate of change of the ln(5x) with respect to x is inversely proportional to x. This property of logarithmic functions can be useful in many applications, such as modeling population growth or analyzing financial data.

## How do you prove the derivative of ln5x?

There are numerous ways to derive differentiation of ln5x. Therefore, we can prove the derivative of ln5 by using;

1. First Principle

2. Implicit Differentiation

3. Product Rule

## Derivative of ln 5x by first principle

The first principle of differentiation tells us that to find the derivative of ln(5x), we can use algebra to find a general expression for the slope of the curve at any point. This method is also called the delta method. The derivative measures the instantaneous rate of change of a function with respect to its independent variable. For ln(5x), the derivative is equal to the reciprocal of x, which represents the slope of the tangent line to the curve at any given point. Mathematically,

f'(x)=lim f(x+h)-f(x)/h

This formula can be applied in a variety of calculus problems, including optimization and related rates problems. Understanding the first principle of differentiation is important because it provides a fundamental concept for calculating the derivative of any function.

## Proof of derivative of ln(5x) by first principle

To differentiate ln(5x) by using first principle, we start by replacing f(x) by ln 5x. Moreover, if you wan to calculate ln6x derivative, simply replace f(x) by ln6x

f'(x)=lim{ln5(x+h)-ln 5x/h}

By logarithmic properties,

f'(x)=lim {ln(x+h/x)/h}

Simplifying,

f'(x)=lim {ln(1+h/x)/h}

Suppose t=hx and h=xt. When h approaches zero, t will also approach zero.

f'(x)=lim {ln(1+t)/xt}

And,

f'(x)=lim ln (1/xt) ln (1+t)

By logarithmic properties, we can write the above equation as,

f'(x)=(1/x) lim ln(1+t)^1/t

Hence by limit formula, we know that,

lim ln(1+t)^1/t =ln e =1

Therefore, the ln5x derivative is;

f.(x)=1/x

## Derivative of ln5x using implicit differentiation

Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly by an equation involving two or more variables. We can use this method to prove the differentiation of ln(5x).

## Proof of derivative of ln 5x by implicit differentiation

We can easily differentiate ln(5x) by using implicit differentiation. So, to prove the derivative of natural log, we can write it as,

y=ln 5x

Converting in exponential form,

e^y=5x

Applying derivative on both sides,

d/dx(e^y)=d/dx(5x)

e^y.dy/dx=5

Now,

dy/dx=5/e^y

Since x=e^y

Therefore,

dy/dx=1/x

Thus, we have shown that the ln(5x) derivative with respect to x is equal to 1/x, which confirms the result we obtained using the first principle of differentiation. Also use the implicit differentiation calculator to find the derivative of a function which is defined implicitly.

## Derivative of ln 5x using product rule

The product rule is a rule in calculus that is used to find the derivative of a product of two functions. This rule can also be used to calculate the ln 5x derivative. The product rule for two functions calculator formula f(x) and g(x) is written as;

d/dx{f(x)g(x)}=f(x)g(x)+g(x)f(x)

The formula of product rule is a fundamental formula in calculus and is used to find the derivatives of many functions, including polynomial functions, trigonometric functions, and exponential functions.

## Proof of ln5x derivative by product rule

We start by assuming that the function ln 5x can be written as;

f(x)= 1. ln 5x

Applying derivative with respect to x,

f(x)=(1. ln 5x)

Applying product rule differentiation,

f(x)=1.(ln 5x)+ln 5x (0)

f(x)=1.(1/x)+0

Therefore,

f(x)=1/x

Hence the derivative of ln5x is always equal to the reciprocal of x.

## How to find the differentiation of ln5x with a calculator?

The easiest way to calculate the d/dx ln5x is by using an online tool. You can use our differential calculator for this. Here, we provide you a step-by-step way to differentiate ln(5x) by using this tool.

1. Write the function as ln x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the d/dx ln5x.

2. Now, select the variable by which you want to differentiate ln5x. Here you have to choose x.

3. Select how many times you want to differentiate ln(5x). In this step, you can choose 2 to find second derivative, 3 to find third derivative and so on.

4. Click on the calculate button.

After completing these steps, you will receive the ln(5x) derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

### What is derivative of ln 2x?

The derivative of ln2x is always equal to 1/x. The derivative ln x is written as;

d / dx (ln 2x) = 1 / x

### What is the derivative of ln 5x?

The derivative of ln(5x) can be calculated as;

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

### What is the derivative of ln 2?

The derivative of ln 2 is zero. It is because ln 2 is a constant number. Therefore, in calculus, the derivative of any constant number is always zero.

### What is derivative used for?

Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. The equation of tangent and normal line to a curve of a function can be calculated by using the derivatives.