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Derivative of Square Root

Complete guide with formula, proof, examples, and graph.

Quick Answer

ddxx=12x\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}

The derivative of \sqrt{x} is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Rewrite the square root using exponent notation.

x=x1/2\sqrt{x} = x^{1/2}

Apply the power rule for differentiation.

Applypowerrule:ddx[xn]=nxn−1Apply power rule: \frac{d}{dx}[x^n] = nx^{n-1}

Substitute n = 1/2 and simplify the exponent.

ddxx1/2=12x1/2−1=12x−1/2\frac{d}{dx}x^{1/2} = \frac{1}{2}x^{1/2 - 1} = \frac{1}{2}x^{-1/2}

Convert back to radical notation for the final answer.

=12x= \frac{1}{2\sqrt{x}}

Graph

Visualization of Square Root and its derivative.

f(x) = \sqrt{x}

f(x)=xf(x) = \sqrt{x}

f'(x) = \frac{1}{2\sqrt{x}}

f′(x)=12xf'(x) = \frac{1}{2\sqrt{x}}
Domain: [0,+∞)[0, +\infty)Range: [0,+∞)[0, +\infty)

Worked Examples

Step-by-step solutions using the chain rule and other techniques.

Find: ddx3x+1\frac{d}{dx}\sqrt{3x+1}

Solution: 323x+1\frac{3}{2\sqrt{3x+1}}

1.Rewrite:(3x+1)1/2,Chainrule:u=3x+1,u′=3Rewrite: (3x+1)^{1/2}, Chain rule: u = 3x+1, u' = 3
2.=12(3x+1)−1/2⋅3=323x+1= \frac{1}{2}(3x+1)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x+1}}

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Related Derivatives

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Frequently Asked Questions

What is the derivative of Square Root?

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The derivative of x\sqrt{x} is 12x\frac{1}{2\sqrt{x}}. This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Square Root?

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The proof uses the limit definition of the derivative. See the Proof section above for the complete step-by-step derivation.

Is the derivative of Square Root always the same?

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Yes, the derivative formula for x\sqrt{x} is constant — it does not depend on x. However, when composed with inner functions (e.g., x\sqrt{x} of u(x)), the chain rule applies.

Where is the derivative of Square Root undefined?

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The derivative is undefined where the original function is not differentiable. Check the domain section for details.

Why is the derivative of Square Root important?

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This derivative appears frequently in physics (wave motion), engineering (signal processing), economics (oscillating models), and many other fields involving periodic or growth phenomena.