This learning article is currently available in English. It is not published as a translated page.

Core derivative rules

Power Rule for Derivatives: Exponents and Conditions

The power rule lowers a fixed exponent by one and uses the old exponent as a coefficient. It does not directly apply when the variable is in the exponent.

Direct answer

d/dx xⁿ = nxⁿ⁻¹

The real interval must support the original power; a composite base also requires the chain rule.

When to use power rule for derivatives

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

Confirm the exponent is fixed and the variable belongs to the base.

Ask first 2

Which condition can change the answer?

What to do

The real interval must support the original power; a composite base also requires the chain rule.

Ask first 3

How should the result be checked?

What to do

The new exponent must be one less and every inner function must contribute its derivative.

Integer powers from the binomial theorem

Expand (x+h)ⁿ and subtract xⁿ.

After division by h, the leading term is nxⁿ⁻¹.

All terms with a remaining h vanish as h→0.

For [g(x)]ⁿ, multiply by g′(x) and retain all domain restrictions.

1
(x+h)ⁿ=xⁿ+nxⁿ⁻¹h+…
2
[(x+h)ⁿ−xⁿ]/h=nxⁿ⁻¹+O(h)
3
[g(x)ⁿ]'=ng(x)ⁿ⁻¹g'(x)

Worked examples

ddx[x5]\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{5}\right]
5 x45\,x^{4}

Lower 5 to 4.

ddx[x−2]\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{-2}\right]
−2 x−3-2\,x^{-3}

Valid where x≠0.

ddx[(x2+1)4]\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^{2} + 1\right)^{4}\right]
8 x (x2+1)38\,x\,\left(x^{2} + 1\right)^{3}

Power rule plus chain rule.

Common mistakes

  • • Forgetting the exponent coefficient.
  • • Using the rule on aˣ.
  • • Omitting the inner derivative.
  • • Losing domain restrictions.

Error clinic: locate the first invalid step

Incorrect attempt

[x⁵]'=x⁴

Why it fails

The original exponent must become a coefficient.

Correction

[x⁵]'=5x⁴

Incorrect attempt

[(x²+1)⁴]'=4(x²+1)³

Why it fails

The inner derivative 2x is missing.

Correction

[(x²+1)⁴]'=8x(x²+1)³

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[x−2]\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{-2}\right]

Check before solving

Valid where x≠0.

Reveal the reference answer

Reference answer

−2 x−3-2\,x^{-3}
ddx[(x2+1)4]\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^{2} + 1\right)^{4}\right]

Check before solving

Power rule plus chain rule.

Reveal the reference answer

Reference answer

8 x (x2+1)38\,x\,\left(x^{2} + 1\right)^{3}

Try the worked examples in the calculator

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

Derivadas:x5x^{5}Derivadas:(x2+1)4\left(x^{2} + 1\right)^{4}
Open the calculator without a preset: Derivadas

Frequently asked questions

Does it work for negative exponents?+

Yes on intervals where the original expression is defined.

Is xˣ a power-rule problem?+

Not directly; logarithmic differentiation handles a varying base and exponent.

Why does a constant differentiate to zero?+

It is a constant multiple of x⁰, whose derivative coefficient is zero.

Conclusion

Use the power rule for a fixed exponent, then add chain-rule and domain checks where needed.

Compare usage plans