Derivative of xlnx

Learn what is the derivative of xlnx and how to calculate it using product rule and first principle of differentiation.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of xlnx

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

What is the derivative of xlnx?

The derivative of xln(x) is calculated by taking the differential of the function with respect to x. This can be written as d/dx (xln(x)). The resulting value is 1 + ln(x). This represents the rate of change of the logarithmic function xln(x). In other words, it tells us how quickly the value of the function is changing as x changes. Knowing the derivative of xln(x) can be useful in a variety of contexts, from calculating slopes and rates of change to solving optimization problems.

Differentiation of xlnx formula

The formula of derivative of x lnx is equal to the sum of 1 and logarithmic function lnx, that is;

d / dx (xlnx) = 1+lnx

This equation expresses the rate of change of the logarithmic function xln(x) with respect to x. By understanding this formula, you can calculate slopes and rates of change, solve optimization problems, and more.

How do you prove the xlnx derivative?

There are various ways to prove the differentiation of xlnx. These are;

  1. First Principle

  2. Product Rule

Each method provids a different way to compute the xlnx differentiation. By using these methods, we can mathematically prove the formula for finding differential of xlnx.

Derivative of xlnx by first principle

According to the first principle of derivative calculator, the xlnx derivative is equal to 1+ln x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

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

This formula allow us to determine the rate of change of a function at a specific point by using limit definition of derivative.

Proof of xlnx differentiation by first principle

To prove the derivative of x ln(x) by using first principle, replace f(x) by xlnx or replace by n x to find the derivative of ln x. f(x)=limh→0f(x+h)-f(x)/h

d(xlnx)/dx = lim [(x+h) ln(x+h) - xlnx]/[(x+h) - x]

= limh→0[x ln(x+h) + h ln(x+h) - xlnx] / h

= limh→0[x ln(x+h) - x lnx + h ln(x+h)] / h

= limh→0[x(ln(x+h) - lnx) + h ln(x+h)] / h

= limh→0[x ln [(x+h)/x] + h ln(x+h)] / h

= limh→0[x ln (1+h/x) + h ln(x+h)] / h

= limh→0 [x ln (1+h/x)] / h + limh→0 ln(x+h)

= limh→0 [ ln (1 + h/x) ] / (h/x) + limh→0 ln(x+h)

Using limit formula lim [ ln (1 + x) ] / x = 1, we get

= 1 + lnx

Hence, we have proved that the formula for the xlnx derivative is equal to 1 + lnx.

Differentiation of xlnx by product rule

The derivative of x ln(x) can be calculated by using product rule formula because an algebraic function can be written as the combination of two functions. The product rule of derivatives is defined as;

[uv] = u.v +u.v

This formula allows you to find the rate of change of a product of two functions at the same time. 

Proof of differential of xlnx by product rule

To differentiate x ln x by using product rule, we start by assuming that,

f(x) = xlnx

By using product rule of differentiation,

f(x) = x.(lnx)+lnx. (x)

We get,

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


f(x) = 1 + lnx

Which is the derivative xlnx.

You can also find derivative using product rule calculator on this website for learning and practice.

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

The easiest way to calculate the differential of xlnx is by using an online tool. You can use our differentiation calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as xlnx in the enter function box. In this step, you need to provide input value as a function as you have to calculate the xlnx differentiation.

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

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

  4. Click on the calculate button.

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

Frequently Asked Questions

How do you drive 2lnx?

The derivative of 2lnx can be simply calculated by applying derivative formula. The formula of derivative of 2lnx is;

d / dx (2lnx) = 2 / x

What is derivative of sin x is cos x?

If we look at the sin x and cos x graphs, we can see that cos x is 0 at all points where sin x is either maximum or minimum. This means that at all critical points of sin x, cos x = 0. Therefore, the derivative of sin x is cos x.

What is the differentiation of xcos x?

The derivative of xcos x can be calculated by using product rule of differentiation. To apply product rule, follow the given steps;

d / dx (xcos x) = (x) cos x + x (cos x)

d / dx (xcos x) = cos x - x.sin x

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