Derivative of Root X
2026-02-28 01:34 Diff
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Last updated on September 30, 2025

The derivative of √x, given by 1/2√x, shows how the square root curve changes with the change in value of x. This concept is used in various fields like physics, engineering, and economics, where the change of growth patterns are analyzed. Now, let us learn about the derivatives of √x and its applications.

What is the Derivative of Root X?

Square root differentiation is the process of finding the derivative of square root functions. The derivative of √x is represented by d/dx(√x) or d/dx(√x½). It involves the process of finding the change of the square root function in respect to the change of the fundamental variable. For √x, the derivative is 1/2√x which represents how the curve of the square root function changes as the value of x changes.

Derivative of Root X Formula

To find the derivative of root x the formula is given below:

             d/dx (√x) = 1/2√x.

Proofs of the Derivative of Root X

The derivative of root x can be proved using various approaches like using the definition of a derivative (first principle rule), using the chain rule, and using the product rule. Let us now see how the various rules are used in detail:

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By First Principle Rule

The step-by-step way to find the derivative of root x using the first principle rule is given below:

Step 1: Recall the First Principle Formula:

The derivative of a function f(x) using the first principle rule is given by

              f’(x) = h→0 f(x + h) - f(x)/h

 Here, h is the small change in x.

Step 2: Substitute f(x) = √x into the formula:

For f(x) = √x substitute this into the formula:


f’(x) = h→0 √x + h  -  x/h

Step 3: Simplify the Numerator:

The numerator √x + h  - √x contains square roots, so multiply and divide by the    conjugate to eliminate the square roots:

√x + h  -  √x/h x √x + h  +  √x/√x + h  +  √x = (x + h) - x/h(√x + h + √x

          Simplify the numerator: 

          (x + h) - x = h.

Step 5: Take the Limit as h → 0:

As h → 0, √x + h → √x. Substitute this into the equation:

f’(x) = 1/√x + √x  = 1/2√x

Final Answer: 

The derivative of the root of x using the first principle is:

          f’(x) = 1/2√x

Using the Chain Rule

To find the Formula of derivative of the root of x using the chain rule, we follow the steps given below:

Step 1: Rewrite x  in Exponent Form:
The root of x can be written as:

                     x = x1/2.

Step 2: Recognize the Chain Rule:
The chain rule states: 

            ddx[f(g(x))] = f’(g(x)) x g’(x).

    Here, we apply this rule to functions that can be expressed in a composite form

    Step 3: Apply the chain rule to x1/2
In the case of x = x1/2, there is no inner function to simplify further, since g(x) = x is the simplest form. Hence, we directly apply the power rule, which is used in special cases of chain rule.

Step 4: Differentiate using the power rule:
The power rule states: 

            ddx[xn] = n x xn-1

Substitute n = ½ into the power rule:

ddx[x1/2] = ½ x x1/2 - 1 

    

    Step 5: Simplify the Exponent:

                             ½ x x1/2 - 1 = ½ x x-1/2

Step 6: Rewrite in the terms of root:
Using the property x-½     = 1x, we can write it as:

                     ½ x x-½  = 12x

    Final Answer:
    The derivative of x is:

                     ddx(x) = 12x

Using the Product Rule

The product rule is used to differentiate a function that is a product of two different functions. The product rule states that:

            ddx[u(x) x v(x)] = u’(x)v(x) + u(x)v’(x).

To find the derivative of the root of x using the product rule, the step-by-step procedure is given below:

Step 1: Rewrite x as a product:
Recall that x = x1/2 we can express this as a product:

                     x = x1/2  = x1/2 x 1

Here, we consider u(x) = x1/2 and v(x) = 1. This will allow us to apply the product rule.

Step 2: Identify u(x) and v(x):
    Let us take:

                         u(x) = x1/2 and v(x) = 1

Step 3: Differentiate u(x) and v(x):
The derivative of u(x) = x1/2 is:

                 u’(x) = 12x-1/2 = 12x

The derivative of v(x) = 1 is:

                 v’(x) = 0

    Step 4: Apply the product rule:
    Substitute the values to the given formula

                         ddx[u(x) x v(x)] = u’(x)v(x) + u(x)v’(x).

         This becomes:

                         ddx[x] = (12x)(1) + (x1/2)(0)


    Step 5: Simplify the Equation:
    The second term becomes 0 as v’(x) = 0. Hence;
                ddx[x] = 12x

    Final Answer:
    The derivative of the root of x using the product rule is:

                ddx[x] = 12x

Higher-Order Derivatives of the Root of X

The higher-order derivatives of the root of x involves finding the derivative of its first derivative and contributing to differentiate successively to obtain a more complex equation. These higher derivatives provide insights into the behavior of the root function and are useful in various applications like mathematical modeling and series expansions.

Nth Derivative of the Root of X

To find the Nth derivative of the root of x, we express it by finding a more convenient form of differentiation. The root or x can be written as x1/2. The general formula used to find the Nth derivative of the root of x is given below

        f(N)(x) = (-1)N-1  * (2N-3)!!2N x x(1/2) - N
Where (2N - 3)!! Is double factorial.

Special Cases:

For N = 0:
The 0th derivative is just the original function of x1/2.
For N = 1:
The first derivative is 12x-1/2
For higher derivatives:
Each differentiation reduces the exponent by 1. As N increases, the coefficients become more complex.
When x = 0:
The Nth derivative for x1/2 at x = 0 is undefined.

Common Mistakes and How to Avoid them in Derivatives of Root X

Problem 1

Find the derivative of √x

Okay, lets begin

The derivative of √x = 1/2√x

Explanation

 Using the power rule:

√x can be written as x1/2

Derivative of xn = n × x(n-1)

Derivative of √x = ½ × x(1/2 - 1)

                       = ½ × x(-1/2)

                       = 1/2√x

Well explained 👍

Problem 2

Find the derivative of √(2x + 1)

Okay, lets begin

The derivative of  √(2x + 1) = 1/√(2x+1)

Explanation

Using the chain rule:

Let u = 2x + 1

Then the function becomes √u

Derivative of √u = 1/2√u

Derivative of u (2x + 1) with respect to x is 2

By the chain rule, the derivative of √(2x + 1) = (1/(2√u)) x 2

=  1/√(2x+1)

Well explained 👍

Problem 3

Find the derivative of √(x² + 3x)

Okay, lets begin

The derivative of √(x2 + 3x) = (2x + 3)/(2(√x2 + 3x))

Explanation

Using the chain rule:

Let u = x2 + 3x

Then the function becomes √u

Derivative of √u = 1/2√u

Derivative of u (x2 + 3x) with respect to x is 2x + 3

By the chain rule, the derivative of √(x2 + 3x) =  (1/(2√u)) x (2x + 3)

=  (2x + 3)/(2√(x2 + 3x))

Well explained 👍

Problem 4

Find the derivative of 1/√x.

Okay, lets begin

The derivative of 1/√x = -1/(2x√x)

Explanation

1/√x can be written as x-1/2 

Derivative of x-1/2 = (-1/2) x x(-1/2 - 1) 

                              =  (-1/2) x x(-3/2)

                              = -1/(2x(3/2))

                              =  -1/(2x√x)

Well explained 👍

Problem 5

Find the derivative of √(sin(x))

Okay, lets begin

The derivative of √(sin(x)) = cos(x)/(2√(sin(x))

Explanation

Using the chain rule:

Let u = sin(x)

Then the function becomes √u

Derivative of √u = 1/2√u

Derivative of u (sin(x)) with respect to x is cos(x)

By the chain rule, the derivative of √(sin(x)) is

√(sin(x)) = 1/2√u x cos(x)

            = cos(x)/(2√(sin(x))

Well explained 👍

FAQs on the Derivative of Root X

1.What is the derivative of √x?

The derivative of √x = 1/2√x

2.What is the domain of the derivative of √x?

The derivative of √x = 1/2√x. Since the denominator cannot be 0, the domain of the derivative is x > 0.

3.What are some applications of the derivative of √x?

The applications of the derivative of x are; finding the rate of change of functions involving square roots, optimizing functions in calculus, solving problems in physics, engineering and other fields.

4.How is the derivative of x related to its slope?

The derivative of √x represents the slope of the tangent line to the curve of function y = √x at any given point.

5.What is the second derivative of √x?

The second derivative of √x = -1/(4x3/2)

Important Glossaries for the Derivative of Root X

  • Derivatives: It is a measure of a function changes as its input changes. In geometry, it is the slope of a tangent line to the functions graph.
  • Slope: A slope is the measure of the steepness of a line. It shows the change in y divided by the change in x.
  • Tangent Line: A tangent line is a line that touches a curve at a single point and has the same slope as the curve at that point.
  • Domain: A domain is the set of all possible input values (x) for with a function is defined.

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