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 differentiation of xlnx 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 xlnx 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;

First Principle

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, 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 e by using first principle, replace f(x) by xlnx or replace by ln x to find the derivative of ln x. f(x)=lim_{h→0}f(x+h)f(x)/h
d(xlnx)/dx = lim [(x+h) ln(x+h)  xlnx]/[(x+h)  x]
= lim_{h→0}[x ln(x+h) + h ln(x+h)  xlnx] / h
= lim_{h→0}[x ln(x+h)  x lnx + h ln(x+h)] / h
= lim_{h→0}[x(ln(x+h)  lnx) + h ln(x+h)] / h
= lim_{h→0}[x ln [(x+h)/x] + h ln(x+h)] / h
= lim_{h→0}[x ln (1+h/x) + h ln(x+h)] / h
= lim_{h→0} [x ln (1+h/x)] / h + lim_{h→0} ln(x+h)
= lim_{h→0} [ ln (1 + h/x) ] / (h/x) + lim_{h→0} ln(x+h)
Introduction to Derivative of sqrt x
Derivatives have a wide range of applications in almost every field of engineering and science. The sqrt x derivative can be calculated by following the rules of differentiation. Or, we can directly find the derivative of the square root of x by applying the first principle of differentiation. In this article, you will learn what the derivative of the square root of x is and how to calculate the derivative by using different approaches.
What is the derivative of square root of x?
The derivative of the square root of x, denoted as d/dx(√x), is a fundamental concept in calculus. It represents the rate of change of the algebraic function x with respect to the variable x and is always equal to 1. The derivative of an algebraic function sqrt(x) helps to calculate the slopes of tangent lines, the velocity of moving objects, and the rate of change of various physical quantities.
Derivative of square root of x formula
The formula of derivative of square root x is equal to 1 / 2√x, that is;
d / dx (√x) = 1 / 2√x
This is a fundamental formula in calculus and is used to calculate the rate of change of the function sqrt x with respect to the variable x.
How do you prove the derivative of the square root of x?
There are various ways to prove the derivative of sqrt(x). These are;

First Principle

Product Rule

Quotient Rule
Each method provids a different way to compute the sqrt(x) differentiation. By using these methods, we can mathematically prove the formula for finding differential of sqrt x.
Derivative of square root of x by first principle
According to the first principle of derivative, the sqrt x derivative is equal to 1 / 2√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 derivative of sqrt(x) by first principle
To prove the √x derivative by using first principle, replace f(x) by √x or replace by x to calculate derivative of x.
f(x) = lim_{h→0}f(x = h)  f(x)/h
f(x) = lim √x = h  √x /h
Rationalizing by multiplying and dividing by √x = h = √x
f(x) = lim (x = h)  x/h[√x = h = √x]
Moreover,
f(x) = lim h/h[√x = h = √x]
And,
f(x) = lim 1/[√x = h = √x]
When h approaches to zero then,
f(x) = 1 / 2√x
Hence the derivative of the square root of x is equal to 1 / 2√x.
Derivative of square root of x by product rule
The derivative of root x can be calculated by using product rule 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
Proof of derivative of square root of x by product rule
To prove the derivative of x by using product rule, we can start by assuming that,
f(x) = 1. √x
By using product rule of differentiation,
f(x) = (1). √x = (√x)
We get,
f(x) = 0 = 1 / 2√x
Hence,
f(x) = 1 / 2√x
Which is the derivative of square root of x.
derivative of sqrt x using quotient rule
Another method for finding the differential of x is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated using the quotient rule. The derivative quotient rule is defined as:
d / dx (f/g) = f(x). g(x)  g(x).f(x) /{g(x)}^{2}
Proof of derivative of √x by quotient rule
To prove the derivative of x, we can write it,
f(x) = √x / 1 = u / v
Supposing that u = √x and v = 1. Now by quotient rule,
f(x) = (vu  uv)/v^{2}
f(x) = [1 d / dx(√x)  √x. d / dx(1)] / (1)2
= [1 / 2√x  √x (0)] / 1
= 1 / 2√x
Hence, we have derived the derivative of sqrt x using the quotient rule of differentiation.
How to find the derivative of √x with a calculator?
The easiest way to calculate the derivative of sqrtx 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.

Write the function as √x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the derivative of √x.

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

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

Click on the calculate button.
After completing these steps, you will receive the derivative of sqrtx within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.
Frequently Asked Questions
What is a square root of x?
The square root of a number x is a number y such that y2 = x. In other words, a number y is equal to x when it is multiplied by y itself. For example, 4 and  4 are the square roots of 16 such that (  4)2 = 16.
What is the derivative of √2?
The derivative of √2 is zero. It is because √2 is a constant number and in calculus, the derivative of a constant is always equal to zero.
What functions cannot be differentiated?
A function is not differentiable at a if its graph contains a vertical tangent line at a. The tangent line to the curve steepens as x approaches a, eventually becoming a vertical line. Because the slope of a vertical line is unknown, the function is not differentiable in this case.