Two steps differentiate any power of $x$ : bring the exponent down in front as a coefficient, then subtract one from it. That is the whole rule — and it is the integration power rule run backwards, with no $+\,C$ .

Theorem The power rule

$\displaystyle \frac{d}{dx}\,x^{n} = n\,x^{\,n-1}$ .

Bring the exponent

n

down to the front. It is now a plain coefficient.

Subtract one from the exponent that is left.

Two steps, in that order, every time. Unlike integration, nothing is added back afterwards — there is no $+\,C$ .

Example Three powers of x

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

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

\dfrac{d}{dx}\,x^{5} = 5\,x^{5-1} = 5x^{4}

The exponent that comes down becomes the coefficient; the exponent that stays is one smaller.

Note The reverse of the integration rule

Differentiating

x^{n}

— multiply by

n

, then subtract one from the exponent. No constant.

Integrating

x^{n}

— add one to the exponent, then divide by the new exponent, and add

C

Each rule undoes the other. Differentiating throws away the constant term, which is why integration has to put a $+\,C$ back.

Concept The derivative is the slope

Differentiation measures the rate of change, so the derivative at each point is the slope of the graph there. The rule and the picture agree.

y = x

has exponent

1

, so

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

. The line

y = x

has slope

1

everywhere — a constant derivative of

1

y = x^{2}

the slope runs from negative on the left, through

0

at the vertex, to positive on the right. A value that goes negative, zero, positive as

x

increases is exactly the line

2x

— and

\dfrac{d}{dx}\,x^{2} = 2x

The same reading works for $y = x^{3}$ : it is always increasing, so its slope stays positive on both sides, dipping to $0$ at the middle. A value that is positive everywhere and $0$ only at the origin is exactly $3x^{2}$ — the parabola that $\dfrac{d}{dx}\,x^{3}$ gives.

Summary

$\dfrac{d}{dx}\,x^{n} = n\,x^{\,n-1}$ : bring the exponent down, subtract one.
$x^{2} \to 2x$ , $\; x^{3} \to 3x^{2}$ , $\; x^{5} \to 5x^{4}$ .
It is the integration power rule backwards, and there is no $+\,C$ .
The derivative is the slope: rising is positive, falling is negative, flat is $0$ .
$y = x$ has slope $1$ everywhere; the traced slope of $x^{2}$ is the line $2x$ , and of $x^{3}$ is the parabola $3x^{2}$ .

The Power Rule for Differentiation