Introduction to the derivative of xlogx

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

What is the derivative of xlog x?

The differentiation of x log x, denoted as d/dx x log x. It represents the rate of change of the natural logarithmic function xlog x with respect to the variable x which is equal to log x+1. The derivative of log x plays a vital role in various fields, including physics, economics, and engineering.

Derivative of xlog x formula

The formula for the differentiation of xlog x is equal to the sum of log x and 1. Mathematically, it can be written as;

$\frac{d}{dx}[x\log x]=\log x + 1$

The above formula represents the rate of change of the natural logarithmic function xlog x with respect to the variable 'x.' We can use various derivative rules such as product rules to derive the xlog x derivative.

How do you prove the differentiation of x log x?

 There are different methods to derive xlog x derivatives. Three commonly used methods are;

1. First Principle
2. Implicit Differentiation
3. Product Rule

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

Derivative of xlog x by first principle

The derivative first principle says that the differentiation of xlogx is equal to the sum of log x and 1. 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,

$f’(x)=\lim_{h\to 0}\frac{f(x+h) – f(x)}{h}$

Proof of derivative of xlogx by first principle

To prove the derivative of xlog x by using the derivative by definition, replace f(x) by xlog x. 

$f'(x)=\lim_{h\to 0}\frac{(x+h)\log(x+h)- x\log x}{h}$

Expanding the logarithm:

$f'(x)=\lim_{h\to 0}\frac{(x+h)(\log x + \log(1+\frac{h}{x})) - x\log x}{h}$

$f’(x)=\lim_{h\to 0}\frac{(x+h)\log x +(x+h)\log(1+\frac{h}{x})- x\log x}{h}$

$f’(x)=\lim_{h\to 0}\frac{(x+h)\log x + (x+h)\left[\frac{h}{x}- \frac{h^2}{2x^2} + O(h^3)\right]-x\log x}{h}$

$f’(x)=\lim_{h\to 0}\frac{(x+h)\log x + \frac{h}{x}\cdot x-\frac{h^2}{2x}+O(h^3)-x\log x}{h}$

$f’(x)=\lim_{h\to 0}\frac{h- \frac{h^2}{2x} + O(h^3) + h\log x}{h} $

$f’(x)=\lim_{h\to 0}\left[1 - \frac{h}{2x} + O(\frac{h^2}{x}) + \log x\right]$

$f’(x) =1-\frac{1}{2x}+O(\frac{h}{x})+\log x$

Now taking the limit as h approaches to 0, we can observe that by using this limit the higher order terms will vanish.

$f'(x) = 1 - (\frac{1}{2x}) + \log x$

Therefore, the derivative of xlog x is;

$f'(x) = 1 - (\frac{1}{2x} + \log x$

Differentiation of xlogx using implicit differentiation 

Implicit function derivative 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 x log x.

Proof of differentiation of xlog x by implicit differentiation 

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

$y = x\log x$

Taking the derivative on the both sides with respect to x, 

$y' = x\frac{d}{dx}\log x + \log x\frac{d}{dx}(x)$

$y' = x(\frac{1}{x}) + \log x(1)$

$y' = 1 + \log x$

Hence, we can also verify the differentiation of xlog x by using implicit differentiation calculator is 1+log x.

Derivative of xlog x using product rule

Another method to find the derivative of xlogx 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 product rule derivative calculator is:

$f'(x) = g(x) · f'(x) + f(x) · g'(x)$

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 xlog x by product rule 

The function xlog x can be written as;

$f(x) = x · \log(x)$

Applying derivative with respect to x,

$\frac{d}{dx}[f(x)]=\frac{d}{dx}[x\log(x)]$

Applying the product rule, 

$f'(x)=g(x) · f'(x) + f(x) · g'(x)$

$f'(x) = log(x)\frac{d}{dx}(x)+x\frac{d}{dx}(\log x)$

Since the derivative of x is 1 and the derivative of log x is 1/x, 

$f'(x)=\log(x).(1) + x\cdot\frac{1}{x}$

$f'(x)= 1+\log x$

Hence the differentiation of xlogx is always equal to 1+log x. The product rule can be used for logarithmic differentiation. Let us verify the xlog x derivative. 

How to find the differentiation of xlog x with a calculator?

The easiest way to calculate the derivative of xlogx 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 xlog 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 xlog x.
2. Now, select the variable by which you want to differentiate x log x. Here you have to choose ‘x’.
3. Select how many times you want to find xlog x differentiation. In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button. After this step, the derivative calculator with steps will provide you the derivatives within a few seconds.

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