AP Calculus AB / Unit 2: Differentiation: Definition and Fundamental Properties

2.5 The Power Rule

The video for this topic is coming soon. The notes and practice below are ready now.

Notes

The rule

$\displaystyle \frac{d}{dx}\,x^n = n\,x^{n-1}$bring the power down in front, then subtract one from the power

Example 1

  1. $\displaystyle f(x) = x^5$the power is 5
  2. $\displaystyle f'(x) = 5x^{4}$bring down the 5, then 5 - 1 = 4

Example 2: a polynomial

  1. $\displaystyle f(x) = 4x^3 - 2x^2 + 7x - 9$do each term on its own
  2. $\displaystyle f'(x) = 12x^2 - 4x + 7$4·3 = 12, 2·2 = 4, 7x → 7, -9 → 0

Example 3: square roots

  1. $\displaystyle f(x) = \sqrt{x} = x^{1/2}$rewrite the root as a power first
  2. $\displaystyle f'(x) = \frac{1}{2}x^{-1/2}$bring down 1/2, then 1/2 - 1 = -1/2
  3. $\displaystyle f'(x) = \frac{1}{2\sqrt{x}}$negative power means it goes to the bottom

Example 4: x on the bottom

  1. $\displaystyle f(x) = \frac{1}{x^3} = x^{-3}$move it up: the power becomes negative
  2. $\displaystyle f'(x) = -3x^{-4}$bring down -3, then -3 - 1 = -4
  3. $\displaystyle f'(x) = -\frac{3}{x^4}$move it back down

Common mistake

$\displaystyle \frac{d}{dx}x^{-3} = -3x^{-2}$
$\displaystyle \frac{d}{dx}x^{-3} = -3x^{-4}$

subtracting 1 from -3 gives -4, not -2

Does it match the graph?

−1−0.50.511.522.532468slope = 2

The power rule

Practice

From the video
If $f(x) = x^4 - 3x$, find $f'(2)$.
Find $\displaystyle \frac{d}{dx}\,x^7$.
Find $\displaystyle \frac{d}{dx}\,\sqrt{x}$.
Find $\displaystyle \frac{d}{dx}\,\frac{1}{x^2}$.
Find $\displaystyle \frac{d}{dx}\,x^{3/2}$.
What is the slope of the tangent line to $y = x^3$ at $x = 2$?
Find $\displaystyle \frac{d}{dx}\,\frac{1}{\sqrt{x}}$.
Find $\displaystyle \frac{d}{dx}\,\pi^2$.

Free response

Write your answer first, then open the scoring guide and check off the points you earned.

Let $f(x) = x^3$.

(a) Find $f'(x)$.

Show the answer and scoring guide

$f'(x) = 3x^2$

Give yourself the points you earned:

(b) Find the slope of the line tangent to the graph of $f$ at $x = -1$.

Show the answer and scoring guide

$f'(-1) = 3(-1)^2 = 3$

Give yourself the points you earned:

(c) Write an equation for the line tangent to the graph of $f$ at $x = -1$.

Show the answer and scoring guide

The point is $(-1, f(-1)) = (-1, -1)$ and the slope is $3$.

$y + 1 = 3(x + 1)$, or $y = 3x + 2$

Give yourself the points you earned: