Derivative of Root x

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, 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, Derivative in Maths

Proof of 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}

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) = √sinx.

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.

Derivative of Root x Practice Questions

Some practice questions on derivative of root x are,

Q1. Find the derivative of the function f(x) = √(3sinx)

Q2. Find the derivative of the function f(x) = √x + 1/√x.

Q3. Find the value of f'(x), if f(x) = x√tanx.

Q4. If y = x√logx, then find the value of dy/dx.

Q5. If y = x/√sinx, find the value of dy/dx.

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 √sinx?

Derivative of √sinx is cosx/2√sinx by using chain rule of differentiation.

What is Derivative of √logx?

Derivative of √logx is 1/2x√logx 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.