What is the Derivative of ln x/x?

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

What is the derivative of (ln x)/x?

The derivative of ln x/x with respect to x is important in calculus and is commonly denoted by d/dx(lnx/x). This derivative can be calculated using the formula 1 - ln(x)/x^2. Essentially, it represents the rate of change of the natural logarithmic function ln x. Another way to represent ln x is as the logarithm of x with base e, which is written as;

$\ln e^x=x$

Since the function y = ln x/x is logarithmic, finding its derivative involves a process called logarithmic differentiation. This technique is particularly useful for functions that involve products, quotients, or powers of logarithmic functions. 

$\ln e^x=x$

Derivative of ln x/x formula

The derivative of ln x/x with respect to x is equal to 1 - ln x/x^2. Mathematically, it can be written as:

$\frac{d}{dx}\left(\frac{\ln x}{x}\right)=\frac{1-\ln x}{x^2}$

This formula is useful in calculus for finding the rate of change of logarithmic functions involving quotients. By applying this formula, we can calculate the slope of the tangent line to the graph of ln x/x at any given point, which is essential in many mathematical applications.

How do you prove the derivative of (ln x)/x?

There are different methods to derive the derivative of ln x/x. Two commonly used methods are;

1. Implicit Differentiation
2. Product Rule

Each method provides a different way to compute the ln x/x derivative. By using these methods, we can mathematically prove the formula for finding the derivative of ln(x)/x.

Derivative of ln x/x 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 x/x.

Proof of derivative of ln x/x by implicit differentiation 

 To prove the derivative of natural log, we can start by writing it as,

$y=\frac{\ln x}{x}$

Rearranging, 

$xy=\ln x$

Applying derivative on both sides,

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

Since the derivative of ln x is 1/x. Now applying product rule on the left side,, 

$x\frac{dy}{dx}+y=\frac{1}{x}$

Now,

$x\frac{dy}{dx}=\frac{1}{x}-y$

$\frac{dy}{dx}=\frac{\frac{1}{x}-y}{x}=\frac{1-xy}{x^2}$

Since y =ln x/x, therefore, the above equation becomes,

$\frac{dy}{dx}=\frac{1-ln x}{x^2}$

You can also use an implicit differentiation calculator to calculate the derivative of ln x/x in just a few steps.

Differentiation of ln x/x using product rule

Another method to find the derivative (ln x)/x is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the derivative product rule calculator is:

d/dx(uv) = u(dv/dx) + (du/dx)v 

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of ln x by product rule 

The function ln x/x can be written as;

$f(x)=\frac{1}{x}\cdot\ln x$

Applying derivative with respect to x,

$f'(x)=\frac{d}{dx}\left(\frac{1}{x}\cdot\ln x\right)$

Applying product rule, 

$f'(x)=frac{1}{x}\frac{d}{dx}(x)+x.\frac{d}{dx}\left(\frac{1}{x}\right)$

$f'(x)=\frac{1}{x}\times\frac{1}{x}+\ln x\times\frac{-1}{x^2}$

Moreover, 

$f'(x)=\frac{1-\ln x}{x^2}$

Hence the derivative of ln x/x can also be verified by using the product rule calculator.

How to find the derivative of ln x/x with a calculator?

The easiest way to calculate the derivative ln x/x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as ln x/x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the differentiation of ln x/x.
2. Now, select the variable by which you want to differentiate ln x/x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate the given function. In this step, you can choose 2 for second, 3 to find the third derivative.
4. Click on the calculate button. After this step, you will get the derivative of ln x/x within a few seconds.