Math Core

Computing every derivative from the limit definition would be slow and error-prone. Fortunately, the definition produces patterns, and those patterns become rules you can apply in one line. The first and most-used rule handles powers of $x$ .

Two rules to start

Constants. The graph of $f(x) = c$ is a horizontal line, so its slope is $0$ everywhere. From the definition, $\dfrac{c - c}{h} = 0$ for every $h$ , so

\frac{d}{dx}[c] = 0

Powers. Look at what the definition gave for small powers:

$f(x)$	$x$	$x^2$	$x^3$	$x^4$
$f'(x)$	$1$	$2x$	$3x^2$	$4x^3$

The exponent comes down in front, and the new exponent is one less.

The power rule

For any real number $n$ ,

\frac{d}{dx}\left[x^n\right] = n x^{n - 1}

This holds for positive integers, negative integers, and fractional exponents, wherever $x^{n-1}$ is defined.

Why it works for whole-number powers

For a positive integer $n$ , expand $(x + h)^n$ with the binomial theorem:

(x + h)^n = x^n + n x^{n - 1} h + \big(\text{terms containing } h^2, h^3, \ldots\big)

Subtract $x^n$ and divide by $h$ :

\frac{(x + h)^n - x^n}{h} = n x^{n - 1} + \big(\text{terms containing } h, h^2, \ldots\big)

As $h \to 0$ , every term that still contains $h$ vanishes, leaving $n x^{n - 1}$ . The proofs for negative and fractional exponents take more work, but the same formula results, and you may use it freely on the AP exam.

Rewrite before you differentiate

The power rule needs the form $x^n$ . Radicals and fractions have to be rewritten with exponents first:

written as	rewrite as
$\sqrt{x}$	$x^{1/2}$
$\sqrt[3]{x^2}$	$x^{2/3}$
$\dfrac{1}{x^4}$	$x^{-4}$
$\dfrac{1}{\sqrt{x}}$	$x^{-1/2}$

After differentiating, you can convert back to radicals and positive exponents if you like. Either form is correct.

Worked example: Integer powers

Differentiate (a) $y = x^7$ and (b) $y = \dfrac{1}{x^3}$ .

(a) $\dfrac{dy}{dx} = 7x^6$ .

(b) Rewrite as $y = x^{-3}$ . Then

\frac{dy}{dx} = -3x^{-3 - 1} = -3x^{-4} = -\frac{3}{x^4}

Note that $-3 - 1 = -4$ : subtracting 1 from a negative exponent makes it more negative.

Worked example: Fractional powers

Differentiate (a) $f(x) = \sqrt{x}$ and (b) $g(x) = \sqrt[3]{x^2}$ .

(a) $f(x) = x^{1/2}$ , so $f'(x) = \dfrac{1}{2} x^{-1/2} = \dfrac{1}{2\sqrt{x}}$ . This matches what the limit definition gave in the previous lesson, with far less work.

(b) $g(x) = x^{2/3}$ , so

g'(x) = \frac{2}{3} x^{2/3 - 1} = \frac{2}{3} x^{-1/3} = \frac{2}{3\sqrt[3]{x}}

Notice that $g'(0)$ is undefined. That's the cusp you saw on the graph of $y = x^{2/3}$ in the differentiability lesson.

Slopes and tangent lines

With the power rule, tangent-line problems become quick.

Worked example: A tangent line in two lines of work

Find the tangent line to $y = x^3$ at $x = 2$ .

The slope is $y'(2) = 3(2)^2 = 12$ , and the point is $(2, 8)$ . The tangent line is

y - 8 = 12(x - 2), \quad \text{or} \quad y = 12x - 16

The tangent line to y = x³ at (2, 8).Open in grapher →

You can also run the question backward: given a slope, find where the curve has it.

Worked example: Where is the slope 12?

At which points on $y = x^3$ is the tangent line parallel to $y = 12x - 16$ ?

Parallel lines have equal slopes, so set $3x^2 = 12$ . Then $x^2 = 4$ and $x = \pm 2$ . The points are $(2, 8)$ and $(-2, -8)$ .

Common mistake

The power rule is for a variable base and a constant exponent. It does not apply to $2^x$ (variable in the exponent) and it gives $0$ , not $n x^{n-1}$ , for a constant like $\pi^3$ or $5^2$ . $\dfrac{d}{dx}[\pi^3] = 0$ because $\pi^3 \approx 31$ is just a number.

Tip

Before you apply the power rule, ask "Is this exactly $x$ to a constant power?" If not, rewrite it until it is. Most power-rule mistakes are rewriting mistakes, such as treating $\dfrac{1}{x^3}$ as $x^3$ or $\sqrt{x}$ as $x^2$ .

Practice

Practice 1

Find $\dfrac{dy}{dx}$ for $y = x^9$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 2

Find $f'(x)$ for $f(x) = \dfrac{1}{x^5}$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 3

Find $f'(x)$ for $f(x) = x\sqrt{x}$ , $x > 0$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 4

Let $f(x) = \sqrt[3]{x}$ . Find $f'(8)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Let $g(x) = \dfrac{1}{\sqrt{x}}$ . Find $g'(4)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Find an equation of the tangent line to $y = x^4$ at $x = -1$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

What is $\dfrac{d}{dx}\left[\pi^3\right]$ ?

Practice 8

At what point on the graph of $y = x^3$ with $x > 0$ is the tangent line parallel to the line $y = 27x + 5$ ?

Enter a point like (2, -3)