Power Rule (Derivative)

\frac{d}{dx}(x^n) = nx^{n-1}

Description

The Power Rule is one of the first and most important techniques you learn in calculus. It provides a quick and easy way to find the derivative of a function where the variable is raised to a constant power, such as $x^2$, $x^{10}$, or even $x^{-3}$.

Before knowing this rule, finding a derivative requires using the formal limit definition, which involves complex algebra and binomial expansion. The Power Rule bypasses all that work with a simple two-step process: 1. Bring the exponent down in front as a multiplier. 2. Subtract one from the original exponent.

This rule works for **any** real number exponent $n$, including: * **Positive Integers:** $x^5 \to 5x^4$ * **Negative Integers:** $x^{-2} \to -2x^{-3}$ * **Fractions (Roots):** $\sqrt{x} = x^{1/2} \to \frac{1}{2}x^{-1/2}$ * **Decimals/Irrationals:** $x^{\pi} \to \pi x^{\pi-1}$

History & Origins

The development of the Power Rule is tied to the birth of calculus in the late 17th century. Isaac Newton (c. 1665): Newton discovered patterns in the derivatives of polynomials while developing his "method of fluxions." He realized that the rate of change of $x^n$ followed a predictable pattern based on the binomial expansion, allowing him to calculate velocities and accelerations without infinite geometric summations. Gottfried Wilhelm Leibniz (c. 1670s): Leibniz, working independently, introduced the $d/dx$ notation we use today. He proved the rule for integer exponents using finite differences and extended it to rational numbers. The rigorous proof for any real number exponent (including irrationals) came much later, requiring the use of logarithmic differentiation and the definitions of exponential functions.

Proof for Positive Integers (Using Limits)

We can prove the rule for any positive integer $n$ using the definition of the derivative and the Binomial Theorem.

Start with the definition of the derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Substitute $f(x) = x^n$: $\lim_{h \to 0} \frac{(x+h)^n - x^n}{h}$

Expand $(x+h)^n$ using the Binomial Theorem: $(x+h)^n = x^n + nx^{n-1}h + \frac{n(n-1)}{2}x^{n-2}h^2 + ... + h^n$

Substitute back into the limit: $\frac{(x^n + nx^{n-1}h + O(h^2)) - x^n}{h}$

Simplify: The $x^n$ terms cancel out: $\frac{nx^{n-1}h + O(h^2)}{h}$

Divide by $h$: $nx^{n-1} + O(h)$ (where $O(h)$ are terms containing $h$)

Take the limit as $h \to 0$: All terms with $h$ vanish, leaving only $nx^{n-1}$.

Variables

Symbol	Meaning
`n`	Power/exponent (Any real number)
`x`	Variable (Base)
`d/dx`	Derivative operator (Rate of change with respect to x)

Examples

Basic Calculation

Problem: Find d/dx(x⁵)

Solution:

5x⁴

Negative Exponents

Problem: Find the derivative of $f(x) = \frac{1}{x^3}$

Solution: -3/x⁴

Rewrite as a power: $\frac{1}{x^3} = x^{-3}$
Identify n: $n = -3$
Apply Power Rule: Bring -3 down, subtract 1 from exponent.
Calculate: $f'(x) = -3x^{-3-1} = -3x^{-4}$
Rewrite without negative exponents: $-\frac{3}{x^4}$

Fractional Exponents (Roots)

Problem: Find the derivative of $f(x) = \sqrt[3]{x^2}$

Solution: 2/(3∛x)

Convert radical to exponent: $\sqrt[3]{x^2} = x^{2/3}$
Identify n: $n = 2/3$
Apply Rule: $f'(x) = \frac{2}{3}x^{2/3 - 1}$
Subtract exponent: $2/3 - 1 = -1/3$
Result: $\frac{2}{3}x^{-1/3}$
Convert back to radical: $\frac{2}{3\sqrt[3]{x}}$

Geometry: Slope of Tangent Line

Problem: Find the slope of the tangent line to the curve $y = x^4$ at the point where $x = 2$.

Solution: m = 32

Find the derivative using Power Rule: $dy/dx = \frac{d}{dx}(x^4) = 4x^3$.
The derivative represents the slope at any x.
Substitute $x = 2$ into the derivative: $m = 4(2)^3$.
Calculate: $m = 4(8) = 32$.
The slope of the tangent line at x=2 is 32.

Common Mistakes

❌ Mistake

Adding instead of subtracting

✅ Correction

Remember the exponent gets SMALLER. $x^5$ becomes $x^4$, not $x^6$. (Integration adds to the power, differentiation subtracts).

❌ Mistake

Handling constants

✅ Correction

The derivative of a constant (like 5 or $\pi$) is 0. Do not treat $5$ as $5x^0$ and try to make it $0x^{-1}$. Just remove it.

❌ Mistake

Negative integers

✅ Correction

$-2 - 1 = -3$, not $-1$. So the derivative of $x^{-2}$ is $-2x^{-3}$, not $-2x^{-1}$.

Real-World Applications

Physics: Motion

The Power Rule is essential for converting Position functions into Velocity and Acceleration. If position is $x(t) = t^3$, then velocity is $v(t) = 3t^2$ and acceleration is $a(t) = 6t$.

Economics: Marginal Analysis

Economists model cost and revenue as polynomial functions. The Power Rule helps calculate "Marginal Cost" (the cost of producing one more unit), which is simply the derivative of the cost function.

Frequently Asked Questions

Does this work for equations like $2^x$?

No! The Power Rule only works when the base is the variable ($x^n$). If the variable is in the exponent ($2^x$), you must use the rules for Exponential Functions.

What if n = 0?

If $n=0$, then $f(x) = x^0 = 1$. The derivative of the constant 1 is 0. The formula gives $0x^{-1} = 0$, so it still works!