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Power Rule

Power Rule

The formula for finding the derivative of a power of a variable.

 

Power Rule formula d/dx x^n = nx^(n-1) with examples: d/dx x^2=8x^2, d/dx x^-5=-5x^-6, d/dx √x=1/(2√x)

 

 

See also

Derivative rules

Key Formula

ddx[xn]=n⋅xn−1\frac{d}{dx}\left[x^n\right] = n \cdot x^{n-1}
Where:
  • xx = The variable being differentiated with respect to
  • nn = The exponent (any real number constant)
  • nxn−1nx^{n-1} = The resulting derivative after applying the rule

Worked Example

Problem: Find the derivative of f(x) = x^5.
Step 1: Identify the exponent n. Here, n = 5.
f(x)=x5f(x) = x^5
Step 2: Bring the exponent down as a coefficient in front of x.
5⋅x ?5 \cdot x^{\,?}
Step 3: Reduce the exponent by 1. The new exponent is 5 − 1 = 4.
f′(x)=5x4f'(x) = 5x^4
Answer: f'(x) = 5x^4

Another Example

This example shows the Power Rule applied to negative and fractional exponents, along with constant coefficients, which is where students most often need practice.

Problem: Find the derivative of g(x) = 4x^{-3} + 7\sqrt{x}.
Step 1: Rewrite the square root using a fractional exponent so the Power Rule can be applied to every term.
g(x)=4x−3+7x1/2g(x) = 4x^{-3} + 7x^{1/2}
Step 2: Apply the Power Rule to the first term. Multiply the coefficient 4 by the exponent −3, then reduce the exponent by 1.
ddx[4x−3]=4⋅(−3) x−3−1=−12x−4\frac{d}{dx}\left[4x^{-3}\right] = 4 \cdot (-3)\,x^{-3-1} = -12x^{-4}
Step 3: Apply the Power Rule to the second term. Multiply the coefficient 7 by the exponent 1/2, then reduce the exponent by 1.
ddx[7x1/2]=7⋅12 x1/2−1=72 x−1/2\frac{d}{dx}\left[7x^{1/2}\right] = 7 \cdot \tfrac{1}{2}\,x^{1/2 - 1} = \tfrac{7}{2}\,x^{-1/2}
Step 4: Combine the results and, if desired, rewrite with positive exponents.
g′(x)=−12x−4+72x−1/2=−12x4+72xg'(x) = -12x^{-4} + \tfrac{7}{2}x^{-1/2} = -\frac{12}{x^4} + \frac{7}{2\sqrt{x}}
Answer: g'(x) = −12x^{−4} + (7/2)x^{−1/2}

Frequently Asked Questions

Does the Power Rule work for negative and fractional exponents?
Yes. The Power Rule works for any real-number exponent, including negative integers and fractions. For example, the derivative of x^{−2} is −2x^{−3}, and the derivative of x^{3/4} is (3/4)x^{−1/4}. Just remember to rewrite roots and reciprocals as powers of x first.
Why doesn't the Power Rule apply to e^x or 2^x?
The Power Rule requires the base to be the variable and the exponent to be a constant. In expressions like e^x or 2^x, the variable is in the exponent, not the base. These are exponential functions and require different differentiation rules. The derivative of e^x is e^x, and the derivative of a^x is a^x ln(a).
What is the derivative of a constant using the Power Rule?
A constant c can be written as c · x^0. Applying the Power Rule gives c · 0 · x^{−1} = 0. This confirms the familiar rule that the derivative of any constant is zero.

Power Rule vs. Chain Rule (with a power)

Power RuleChain Rule (with a power)
DefinitionDerivative of x raised to a constant powerDerivative of a composite function
Formulad/dx [x^n] = n x^{n−1}d/dx [u^n] = n u^{n−1} · u', where u is a function of x
When to useThe base is simply x (or a constant times x^n)The base is a more complex expression, like (3x + 1)^5
Exampled/dx [x^4] = 4x^3d/dx [(2x+1)^4] = 4(2x+1)^3 · 2 = 8(2x+1)^3

Why It Matters

The Power Rule is typically the very first derivative rule students learn in calculus, and it appears in nearly every differentiation problem. Polynomials, which dominate early calculus courses, are differentiated entirely with the Power Rule. It also forms the foundation for more advanced rules—the Product Rule, Quotient Rule, and Chain Rule all frequently use the Power Rule as part of their computation.

Common Mistakes

Mistake: Forgetting to subtract 1 from the exponent after bringing it down.
Correction: The Power Rule has two parts: multiply by the exponent AND reduce it by 1. Writing the derivative of x^3 as 3x^3 instead of 3x^2 is incorrect.
Mistake: Trying to use the Power Rule when the variable is in the exponent, such as differentiating 2^x.
Correction: The Power Rule only applies when the base is the variable and the exponent is a constant (x^n). For a^x, use the exponential derivative rule: d/dx [a^x] = a^x ln(a).

Related Terms

  • Derivative — The Power Rule computes derivatives
  • Derivative Rules — Collection of rules including the Power Rule
  • Power — The exponent that the rule acts on
  • Variable — The base being raised to a power
  • Formula — The Power Rule is a key calculus formula
  • Chain Rule — Extends the Power Rule to composite functions
  • Product Rule — Differentiation rule often combined with Power Rule
  • Polynomial — Differentiated term-by-term using the Power Rule