Derivative of Root x is (1/2)x^-1/2 or 1/(2√x). In general, the derivative of a function is defined as the change in the dependent variable, i.e. y = f(x) with respect to the independent variable, i.e. x. This process is also known as differentiation in calculus. Root x is an abbreviation used for the square root function which is mathematically represented as √x or x^1/2 (x raised to the power half).

In this article, we will discuss the derivative in math, the derivative of root x, various methods to derive it including the first principle method and the power rule, some solved examples, and practice problems.

What is Derivative of Root x?

Derivative of Root x is 1/2√x. Root x is an algebraic function. Thus, change in the root x function with respect to change in x is given as 1/2√x. The formula for the derivative of Root x can be written as follows:

Derivative of Root x Formula

Formula for the derivative of root x is given by the formula,

(d/dx) [√x] = 1/2√x

(√x)’ = 1/2√x

It can be derived using,

• First Principle of Differentiation

• Power Rule

Both of them are discussed as follows

Learn,

• Calculus in Maths
• Derivative in Maths

How to Find Derivative of Root x

There are two methods to find the derivative of root x:

• Using First Principle of Differentiation
• Using Power Rule

Derivative of Root x Using First Principle

First principle of differentiation state that derivative of a function f(x) is defined as,

f'(x) = lim_h→0 [f(x + h) – f(x)]/[(x + h) – x]

f'(x) = lim_h→0 [f(x + h) – f(x)]/ h

Putting f(x) = √x, to find derivative of root x, we get,

f'(x) = lim_h→0 [√(x + h) – √(x)]/ h

Multiplying numerator and denominator by √(x + h) + √(x), we get,

⇒ lim_h→0 [√(x + h) – √(x)]×[√(x + h) + √(x)]/[h×(√(x + h) + √(x))]

⇒ lim_h→0 [|x+h-x|] / [h×(√(x + h) + √(x))]

⇒ lim_h→0 [h] / [h×(√(x + h) + √(x))]

⇒ lim_h→0 1/ [(√(x + h) + √(x))]

⇒ 1/ [(√(x + 0) + √(x))]

⇒ 1/2√x

Hence, we have derived the derivative of root x by using first principle of differentiation.

Derivative of Root x Using Power Rule

Root x is an algebraic function which can be represented as x^1/2. The Power Rule in differentiation states that,

For any function of the form xⁿ, where n is any real number, the derivative of the function is nx^n-1.

Applying the power rule to find derivative of x^1/2, we get,

(x^1/2)’ = 1/2(x)^1/2-1

⇒ 1/2(x)^-1/2

⇒ 1/2x^1/2 or 1/2√x

Thus, we derived the derivative of root x using the Power Rule.

nth Derivative of Root x

nth derivative of root x is finding the derivative of root x successively n times. If we differentiate any function two times successively then it is called second order derivative. In this manner, if we differentiate any function n times successively we call it nth order derivative

let f(x) = √x = x^1/2

⇒ f'(x) = 1/2(x)^{1/2 – 1}

⇒ f”(x) = 1/2(1/2 – 1)(x)^{1/2 – 1 – 1} = 1/2(1/2 – 1)(x)^{1/2 – 2}

⇒ f”'(x) = 1/2(1/2 – 1)(1/2 – 1 – 1)(x)^{1/2 – 1 – 1 – 1} = 1/2(1/2 – 1)(1/2 – 1 – 1)(x)^{1/2 – 3}

Based on the above pattern, the nth derivative of root x is given as

⇒ fⁿ(x) = 1/2(1/2 – 1)……(1/2 – (n – 1))(x)^{1/2 – n}

Read More:

• Derivative of log x
• Derivatives of Logarithmic functions
• Derivative formula for inverse trigonometric function
• Application of Derivatives
• Derivatives
• Algebra of Derivative of Functions

Application of Derivative of Root x

There are following applications of derivative of root x

• Rate of Change: The derivative of root x​ represents the rate of change of the square root function at any given point x. This is particularly useful in optimization problems, where you need to find the maximum or minimum value of a function.
• Curve Sketching: Understanding the derivative helps in sketching the curve of root x​ accurately, including identifying critical points, inflection points, and concavity.
• Kinematics: In physics, the derivative of x​ can be interpreted as the velocity of an object with respect to time. For example, if x represents distance and t represents time, the derivative d​(√x​)/dt gives the velocity of the object.
• Interest Rates: In finance and economics, the derivative of root x​ can be used to calculate the rate of change of interest rates or investment returns over time.

Also, Check

• Derivative Formulas in Calculus
• Derivative of x root(x)
• Derivatives of Polynomial Functions
• Derivatives of Composite Functions

Solved Examples on Derivative of Root x

Examples on Derivative of Root x are added below,

Example 1: Find the derivative of 3√x.

Solution:

Let, y = 3√x,

We know that,

Derivative of √x is 1/2√x. And, (cf(x))’ = cf'(x)

⇒ y’ = 3(1/2√x) = 3/2√x.

Thus, derivative of 3√x comes out to be 3/2√x.

Example 2: Find the derivative of f(x) = √sin x.

Solution:

Here, f(x) = √sin x

To find f'(x), we apply chain rule of differentiation,

⇒ d/dx(√sin x) = (1/2√sinx)*(d/dx(sin x))

⇒ d/dx(√sin x) = (1/2√sinx)*(cos x)

⇒ d/dx(√sin x) = cos x/2√sin x

Thus, derivative of √sin x comes out to be cos x/2√sin x.

Example 3: Find the derivative of the function given by p(x) = (√x + 4)sinx.

Solution:

Here, we see that two functions are given in product, so we apply the product rule to find the derivative of the given function. Thus,

p'(x) = (√x + 4)’sinx + (√x + 4)(sinx)’

⇒ (1/2√x)sinx + (√x + 4)cosx

Thus, we obtain p'(x) = sinx/2√x + (√x)cosx + 4cosx

Example 4: For f(x) = log √x, what is the value of f'(x)?

Solution:

We know that derivative of log x is 1/x and that for √x is 1/2√x.

Thus, by chain rule, for f(x) = log √x, we have,

⇒ f'(x) = 1/√x * d/dx(√x)

⇒ f'(x) = 1/√x * 1/2√x

⇒ f'(x) = 1/2x

Thus, for f(x) = log √x, we get f'(x) = 1/2x.

Example 5: If y = sin√x, what is the value of dy/dx?

Solution:

Here, y = sin√x,

By chain rule, we get,

⇒ dy/dx = cos√x * d/dx(√x)

⇒ dy/dx = cos√x/2√x

Thus, we get dy/dx = cos√x/2√x for y = sin√x.

People Also Read:

• Derivative of Cotx
• Derivative of ln x (natural log)

Derivative of Root x – Practice Questions

Some practice questions on derivative of root x are,

1. Find the derivative of the function f(x) = √(3sinx)

2. Find the derivative of the function f(x) = √x + 1/√x.

3. Find the value of f'(x), if f(x) = x√tanx.

4. If y = x√logx, then find the value of dy/dx.

5. If y = x/√sinx, find the value of dy/dx.

Summary

The derivative of x, denoted as (d/dx) [√x] , measures how the function √x changes with respect to x. The process to find this derivative involves expressing √x as x^1/2 and then applying the power rule of differentiation, which states that the derivative of x^1/2 is 1/2x^-1/2 ,which simplifies to 1/(2√x). Therefore, the derivative of , which simplifies to . This derivative has several practical applications across different fields. In physics, it helps understand motion under gravity where distance is related to the square root of time. In economics, it models diminishing returns where outputs grow at decreasing rates. Engineering applications include analyzing stresses and strains in materials. In biology, it describes relationships like metabolic rate scaling with body size. Additionally, in optimization problems, it aids in finding minimum or maximum values in cost minimization and efficiency maximization scenarios. Understanding this derivative is crucial for precise modeling and analysis in various scientific and engineering disciplines.

Derivative of Root x – FAQs

What is Derivative of a Function?

Derivative of a function implies the change in the functional value with respect to the change in input variable. For physical quantities, derivative gives the rate of change of the quantity with input variables.

What is Derivative of Square Root x?

Derivative of Square Root of x is 1/2√x.

What are Methods to Find Derivative of Root x?

Methods to find the derivative of Root x are as follows:

• First Principle of Differentiation
• Power Rule

What is Derivative of √sin x?

Derivative of √sin x is cos x/2√sin x by using chain rule of differentiation.

What is Derivative of √log x?

Derivative of √log x is 1/2x√log x by using chain rule of differentiation.

What is the Application of Derivative of Root x?

The application of derivative root x is to find the derivative of other algebraic functions where root x is involved.