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Radical derivatives

Derivative of √x: Formula, Domain, and x=0

The square-root function exists at zero, yet its finite ordinary derivative does not. This separates the function domain from the derivative domain.

Direct answer

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

Over the reals √x exists for x≥0, but this finite derivative requires x>0.

When to use derivative of sqrt(x)

Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.

Ask first 1

Which structure controls the method?

What to do

Use power and chain rules, then inspect the radicand and denominator domains.

Ask first 2

Which condition can change the answer?

What to do

Over the reals √x exists for x≥0, but this finite derivative requires x>0.

Ask first 3

How should the result be checked?

What to do

Record where the radicand is defined and where the derivative denominator is nonzero.

Rationalize the difference quotient

Begin with [√(x+h)−√x]/h.

Multiply by the conjugate over itself.

The numerator becomes h and cancels, leaving 1/[√(x+h)+√x].

The limit is 1/(2√x) for x>0; at zero the right-hand slopes have no finite limit.

1
x=x1/2\sqrt{x}=x^{1/2}
2
ddxx=12x−1/2\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x}=\frac{1}{2}x^{-1/2}
3
ddxg(x)=g′(x)2g(x)\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{g(x)}=\frac{g^{\prime}(x)}{2\sqrt{g(x)}}

Worked examples

ddxx\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x}
12x\frac{1}{2\sqrt{x}}

Valid for x>0.

ddxx2+4\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x^2+4}
xx2+4\frac{x}{\sqrt{x^2+4}}

The radicand stays positive.

ddx3x+1\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{3x+1}
323x+1\frac{3}{2\sqrt{3x+1}}

Require 3x+1>0 for a finite derivative.

Common mistakes

  • • Missing 1/2.
  • • Forgetting the radicand derivative.
  • • Claiming the formula is finite at zero.
  • • Replacing √(x²) with x instead of |x|.

Error clinic: locate the first invalid step

Incorrect attempt

[√x]'=1/√x

Why it fails

The coefficient 1/2 is missing.

Correction

1/(2√x)

Incorrect attempt

[√(x²+4)]'=1/[2√(x²+4)]

Why it fails

The radicand derivative 2x is missing.

Correction

xx2+4\frac{x}{\sqrt{x^{2} + 4}}

Practice the same idea with a changed structure

Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.

ddx[x2+4]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sqrt{x^{2} + 4}\right]

Check before solving

The radicand stays positive.

Reveal the reference answer

Reference answer

xx2+4\frac{x}{\sqrt{x^{2} + 4}}
ddx[3 x+1]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sqrt{3\,x + 1}\right]

Check before solving

Require 3x+1>0 for a finite derivative.

Reveal the reference answer

Reference answer

32 3 x+1\frac{3}{2\,\sqrt{3\,x + 1}}

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Derivadas:x\sqrt{x}Derivadas:x2+4\sqrt{x^{2} + 4}Derivadas:3 x+1\sqrt{3\,x + 1}
Open the calculator without a preset: Derivadas

Frequently asked questions

Why not differentiable at zero?+

The right-hand secant slopes grow without a finite bound.

Can a root derivative exist where its radicand is zero?+

Sometimes after simplification, but the direct formula requires separate analysis there.

Is √(x²)=x?+

Over the reals it is |x|, not x for negative inputs.

Conclusion

Treat √x as a power, multiply by the inner derivative, and distinguish x≥0 from the derivative condition x>0.

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