Grade by Grade
A. Rate of Change · Derivatives and Differentiation Rules

The Power Rule

A3.1About 65 minutes
Learning goals
  • I can explain why the derivative of xⁿ is nxⁿ⁻¹ for natural numbers n.
  • I can use the power rule to differentiate powers of x quickly.
  • I can verify the power rule numerically and by comparing tangent slopes on a graph.
  • I can use the power rule to find tangent lines and points with a given slope.

Words to know

power rule
For natural numbers n, d/dx (xⁿ) = nxⁿ⁻¹.
natural number
One of the counting numbers 1, 2, 3, …
d/dx
An instruction meaning "take the derivative with respect to x of".
parallel tangents
Tangent lines with equal slopes.

A shortcut you have already noticed

From first principles you found:

  • d/dx (x²) = 2x
  • d/dx (x³) = 3x²
  • d/dx (x⁴) = 4x³ (a challenge question in The Derivative from First Principles)

Do you see the pattern? The exponent comes down in front, and the new exponent is one less. This is the power rule, the first and most-used rule of differentiation.

The power rule

If f(x) = xⁿ, where n is a natural number, then f′(x) = nxⁿ⁻¹.

In Leibniz notation: d/dx (xⁿ) = nxⁿ⁻¹.

Two special cases: d/dx (x) = 1 (n = 1: the line y = x has slope 1) and d/dx (1) = 0 (think of 1 as x⁰).

Why it works

Use first principles with f(x) = xⁿ:

f′(x) = lim (h → 0) [(x + h)ⁿ − xⁿ] ÷ h

Multiply out (x + h)ⁿ = (x + h)(x + h)…(x + h), with n brackets. Each term of the product comes from choosing either x or h from every bracket.

  • Choose x every time: one term, xⁿ.
  • Choose h exactly once: n ways, each giving xⁿ⁻¹h, for a total of nxⁿ⁻¹h.
  • Choose h two or more times: every such term contains h² (or a higher power).

So (x + h)ⁿ − xⁿ = nxⁿ⁻¹h + (terms with h² or higher). Dividing by h leaves nxⁿ⁻¹ + (terms with at least one h). As h → 0, those terms vanish, leaving nxⁿ⁻¹. (If you study the binomial theorem in Mathematics of Data Management, MDM4U, you will recognize n as the binomial coefficient "n choose 1".)

(x + h)(x + h)(x + h): pick one term from each brackethxx→ x²hxhx→ x²hxxh→ x²h3 ways:3x²hEvery other product has h² or h³, which disappear after dividing by h and letting h → 0.
Why the 3 comes down in front: when you multiply out (x + h)³, the terms with exactly one h come from choosing h in one bracket and x in the other two. There are 3 ways, giving 3x²h. For (x + h)ⁿ there are n ways, giving nxⁿ⁻¹h.
Worked example

Example 1: Using the power rule

Differentiate: (a) y = x⁵ (b) f(x) = x¹² (c) g(t) = t (d) y = x¹⁰⁰

  1. (a) Bring down the 5 and subtract 1 from the exponent: dy/dx = 5x⁴.
  2. (b) f′(x) = 12x¹¹.
  3. (c) g(t) = t¹, so g′(t) = 1t⁰ = 1. (The variable can be any letter; differentiate with respect to it.)
  4. (d) dy/dx = 100x⁹⁹.

(a) 5x⁴ (b) 12x¹¹ (c) 1 (d) 100x⁹⁹. Compare (d) with expanding (x + h)¹⁰⁰ by hand!

Worked example

Example 2: A slope and a tangent line

Find the equation of the tangent to y = x⁵ at the point (−1, −1).

  1. dy/dx = 5x⁴.
  2. At x = −1: slope = 5(−1)⁴ = 5(1) = 5.
  3. Point-slope form: y − (−1) = 5(x − (−1)), so y + 1 = 5(x + 1).
  4. y = 5x + 4.

y = 5x + 4

xy-11246slope 4(1)³ = 4slope 4(−1)³ = −4slope 4(1.5)³ = 13.5slope 0y = x⁴
Checking the power rule on y = x⁴, whose derivative is 4x³. The dashed tangents at x = −1, 1 and 1.5 have slopes −4, 4 and 13.5, and the tangent at the origin is horizontal.
Worked example

Example 3: Where is the slope a given value?

At which points on y = x³ is the tangent slope 12?

  1. dy/dx = 3x².
  2. Set 3x² = 12, so x² = 4 and x = 2 or x = −2.
  3. y-values: 2³ = 8 and (−2)³ = −8.

At (2, 8) and (−2, −8). The two tangents are parallel because they have the same slope.

Worked example

Example 4: Verifying the rule numerically

Check the power rule for f(x) = x⁶ at x = 1.2 using the centred estimate with h = 0.001.

  1. Power rule: f′(1.2) = 6(1.2)⁵ = 6(2.48832) = 14.92992.
  2. Estimate: f(1.201) = 1.201⁶ ≈ 3.0009451, f(1.199) ≈ 2.9710851. Keep all calculator digits: [f(1.201) − f(1.199)] ÷ 0.002 ≈ 14.92995.
  3. The two values differ by less than 0.0001.

The power rule gives 14.92992 and the estimate gives about 14.92995. ✓

f(x)xx²x³x⁴x⁵xⁿ
f′(x)12x3x²4x³5x⁴nxⁿ⁻¹
The power rule for natural-number exponents.
Just for fun

The power rule is so short that students sometimes say it out loud as a chant: "Bring it down, drop it by one." It is one of the few rules in mathematics that fits on a sticky note and still does serious work in engineering and physics.

Hands-on activity

Power-rule detective

You need: a scientific calculator, paper and a pencil.

  1. Pick a function xⁿ with n between 2 and 7, and a value of a between 0.5 and 2.
  2. Use the centred estimate [f(a + 0.001) − f(a − 0.001)] ÷ 0.002. Keep every digit on your calculator until the end.
  3. Now use the power rule: f′(a) = naⁿ⁻¹.
  4. Repeat for two more functions. Then ask a family member to pick n and a, and race: you use the power rule while they use the centred estimate.
  5. Finally, try f(x) = x⁵ at a = 0. What do both methods give?

What to look for: the two methods should agree to at least three decimal places every time (the power rule wins the race easily). For x⁵ at 0, both give 0 (the estimate works out to 0.000000000001, which is 0 apart from a tiny leftover from using h = 0.001): the tangent at the origin is horizontal.

Your turn

1

Differentiate y = x⁷.

2

Differentiate f(x) = x²⁰.

3

Find f′(3) for f(x) = x⁴.

4

Find the slope of the tangent to y = x⁶ at x = −1.

5

Find the equation of the tangent to y = x² at (3, 9).

6

At which point on y = x² is the tangent slope −8?

7

What is d/dx (x)? Explain with a graph.

More practice: warm-up

1

d/dx (x³) = ?

2

d/dx (x⁹) = ?

3

If y = t⁴, find dy/dt.

4

What is the derivative of the constant 1?

5

Find f′(1) for f(x) = x⁸.

More practice: on your own

1

Find the equation of the tangent to y = x⁴ at x = 2.

2

Find all points on y = x⁵ where the tangent slope is 80.

3

Find the point on y = x³ where the tangent is parallel to y = 27x + 1, with x > 0.

4

Is there any point on y = x⁴ where the tangent slope is the same as at x = 1? Explain.

5

For y = x³, show that the tangent slope is never negative. What does this tell you about the graph?

Challenge

1

The tangent to y = x³ at x = a meets the curve again at another point. Find that point in terms of a (a ≠ 0). Hint: x³ − a³ = (x − a)(x² + ax + a²).

2

Find the value of n if the slope of y = xⁿ at x = 2 is 80.

3

Show that the tangents to y = x² at x = a and x = −a meet on the y-axis, and find where.

Check your understanding

Quiz

1.

What is d/dx (x⁸)?

2.

For f(x) = x³, what is f′(−2)?

3.

Which statement of the power rule is correct for natural numbers n?

4.

The tangent to y = x⁴ at x = 1 has equation:

5.

At which x-values does y = x³ have tangent slope 3?

6.

In the proof of the power rule, why do the terms with h² disappear?

For parents and teachers

Big idea (A3.1): for any natural number n, the derivative of xⁿ is nxⁿ⁻¹. This replaces long first-principles calculations with one line.

How to help: Call out powers ("x to the 7") and have your teen answer with the derivative ("7x to the 6"). Then ask for a slope at a point, such as the slope of y = x³ at x = 2 (answer: 12).

Common difficulties:

  • Forgetting to subtract 1 from the exponent, or adding 1 instead.
  • Mixing up f′(a) (the slope) with f(a) (the y-value) when writing a tangent line.
  • Losing negative signs with odd powers: (−1)⁵ = −1 but (−1)⁴ = 1.
  • Thinking the derivative of x is x or 0; it is 1.