A negative exponent, a fraction, a bare constant — the power rule $\dfrac{d}{dx}\,x^{n} = n\,x^{\,n-1}$ covers all of them. Here are five cases that look like exceptions, each run through the same two steps: bring the exponent down, subtract one.

Case 1

y = x

Write $x$ with its exponent shown: $x = x^{1}$ . Then apply the rule.

\dfrac{d}{dx}\,x^{1} = 1\cdot x^{1-1} = 1\cdot x^{0} = 1

$x^{0} = 1$ , so the derivative is the constant $1$ — the line $y = x$ has slope $1$ everywhere.

Case 2

y = 5

, a constant

A constant never changes, so its rate of change is

0

. The rule gives the same answer: write

5

5x^{0}

, and differentiating multiplies by that exponent.

\dfrac{d}{dx}\,5x^{0} = 0\cdot 5\,x^{0-1} = 0\cdot 5x^{-1} = 0

The coefficient of $0$ out front kills the whole term, whatever the exponent does. The derivative of any constant is $0$ .

Case 3

y = 4x^{3}

, a coefficient out front

A constant multiplier just rides along. Differentiate the power and keep the $4$ attached.

\dfrac{d}{dx}\,4x^{3} = 4\cdot 3\,x^{3-1} = 12x^{2}

The exponent $3$ comes down and multiplies the $4$ to give $12$ ; the exponent left behind is $2$ .

Case 4

y = x^{-2}

, a negative exponent

Nothing changes. Bring $-2$ down, then subtract one from it: $-2 - 1 = -3$ .

\dfrac{d}{dx}\,x^{-2} = -2\,x^{-2-1} = -2x^{-3}

Watch the signs: subtracting one from a negative exponent makes it more negative. Written as a fraction, $-2x^{-3} = -\dfrac{2}{x^{3}}$ .

Case 5

y = \sqrt{x}

, a fractional exponent

A root is a fractional exponent: $\sqrt{x} = x^{1/2}$ . Bring $\tfrac{1}{2}$ down, then $\tfrac{1}{2} - 1 = -\tfrac{1}{2}$ .

\dfrac{d}{dx}\,x^{1/2} = \dfrac{1}{2}\,x^{-1/2} = \dfrac{1}{2\sqrt{x}}

The negative exponent moves $x$ into the denominator, and $x^{1/2}$ there is written as $\sqrt{x}$ .

Summary

$x = x^{1}$ , so $\dfrac{d}{dx}\,x = 1$ .
A constant is $c\,x^{0}$ ; the rule multiplies by $0$ , so the derivative is $0$ .
$\dfrac{d}{dx}\,4x^{3} = 12x^{2}$ — the coefficient rides along.
$\dfrac{d}{dx}\,x^{-2} = -2x^{-3}$ — subtract one from $-2$ to get $-3$ .
$\dfrac{d}{dx}\,\sqrt{x} = \dfrac{1}{2\sqrt{x}}$ — $\sqrt{x}$ is $x^{1/2}$ .

The Power Rule — Special Cases