# Derivative of Square root function

## Formula

$\dfrac{d}{dx}{(\sqrt{x})}$ $\,=\,$ $\dfrac{1}{2\sqrt{x}}$

The derivative of square root of a variable with respect to same variable is equal to quotient of one by two times the square root of variable, is called the derivative rule for square root of a variable.

### Introduction

$x$ is a variable and square root of this variable is $\sqrt{x}$. The derivative of the square root of $x$ with respect to $x$ is written in mathematics as follows.

$\dfrac{d}{dx}{(\sqrt{x})}$

The differentiation of $\sqrt{x}$ with respect to $x$ is equal to the ratio of one to two times square root of $x$.

$\dfrac{d}{dx}{(\sqrt{x})}$ $\,=\,$ $\dfrac{1}{2\sqrt{x}}$

This differentiation rule is used as formula in differential calculus to find the derivative of square root of any variable. The derivative of square root of variable formula can be written in terms of any variable.

$(1) \,\,\,$ $\dfrac{d}{dm}{(\sqrt{m})}$ $\,=\,$ $\dfrac{1}{2\sqrt{m}}$

$(2) \,\,\,$ $\dfrac{d}{dp}{(\sqrt{p})}$ $\,=\,$ $\dfrac{1}{2\sqrt{p}}$

$(3) \,\,\,$ $\dfrac{d}{dy}{(\sqrt{y})}$ $\,=\,$ $\dfrac{1}{2\sqrt{y}}$

### Proof

Learn how to prove the differentiation of square root of $x$ with respect to $x$ formula in differential calculus from first principle.

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