AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/2/2-5)
Unit 2 · Topic 2.5
2.5 Applying the Power Rule
The power rule gives the derivative of xⁿ in one step: bring the exponent down and lower it by one, so d/dx xⁿ = n·xⁿ⁻¹. It works for any real exponent, so rewrite roots and fractions as powers before using it.
Key terms
- power rule
- exponent
- rewriting radicals as powers
- negative exponents
The rule
For any real number n, d/dx (xⁿ) = n·xⁿ⁻¹, wherever xⁿ⁻¹ is defined.
Examples: d/dx x⁵ = 5x⁴. d/dx x = 1 (since x = x¹, the derivative is 1·x⁰ = 1). d/dx x⁻³ = −3x⁻⁴. d/dx x^(1/2) = ½·x^(−1/2).
Once you have the derivative, you can find the slope at any x. To find where the slope equals a given number, set the derivative equal to that number and solve. For y = x³, y′ = 3x² equals 12 when x = 2 or x = −2, so there are two points on the curve with slope 12.
Fractional exponents are written here as x^(1/2), x^(2/3) and so on, the way you'd type them into a calculator. x^(1/2) means √x and x^(2/3) means ∛(x²).
Rewrite first
The power rule only works on the form xⁿ. Before differentiating, rewrite roots as fractional exponents and fractions with x in the bottom as negative exponents:
| Original | Rewritten | Derivative |
|---|---|---|
| √x | x^(1/2) | ½x^(−1/2) = 1/(2√x) |
| ∛(x²) | x^(2/3) | (2/3)x^(−1/3) |
| 1/x | x⁻¹ | −x⁻² = −1/x² |
| 1/x³ | x⁻³ | −3x⁻⁴ |
| 1/√x | x^(−1/2) | −½x^(−3/2) |
Coefficients
A constant coefficient just multiplies the result (more on that in 2.6). Some examples worth checking yourself:
- 6x^(4/3) → 6·(4/3)x^(1/3) = 8x^(1/3)
- 5/x² = 5x⁻² → −10x⁻³ = −10/x³
- 4√x = 4x^(1/2) → 2x^(−1/2) = 2/√x
- x/∛x = x^(2/3) → (2/3)x^(−1/3)
Why it works for positive whole numbers
For f(x) = xⁿ with n a positive integer, expand (x + h)ⁿ = xⁿ + n·xⁿ⁻¹h + (terms with h² or higher). In the difference quotient ((x + h)ⁿ − xⁿ)/h, the xⁿ cancels, leaving n·xⁿ⁻¹ + (terms with h). As h→0, only n·xⁿ⁻¹ survives. The rule holds for all real exponents too, which you can take as given.
Constants that look like powers
π³, e² and 5⁴ are constants, not powers of x. Their derivatives are 0. Only the variable gets the power rule. Also watch for coefficients in the bottom: 1/(3x²) means (1/3)x⁻², not 3x⁻².
Where it fails
The derivative only exists where the new expression is defined. For x^(2/3), the derivative (2/3)x^(−1/3) is undefined at x = 0. That's the cusp you met in 2.4. For ∛x, the derivative (1/3)x^(−2/3) is undefined at 0, giving a vertical tangent.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Rewriting before using the power rule
Differentiate (a) y = ∛(x²) and (b) y = 1/√x.
Show the solutionHide the solution
- Step 1: (a) Rewrite: y = x^(2/3). Power rule: y′ = (2/3)x^(2/3 − 1) = (2/3)x^(−1/3).
- Step 2: Written with a root: y′ = 2 / (3∛x).
- Step 3: (b) Rewrite: y = x^(−1/2). Power rule: y′ = (−1/2)x^(−3/2).
- Step 4: Written with a root: y′ = −1 / (2x√x).
Answer: (a) y′ = (2/3)x^(−1/3) = 2/(3∛x); (b) y′ = −½x^(−3/2) = −1/(2x√x)
- Example 2
Tangent line to y = √x
Find the equation of the line tangent to y = √x at x = 4.
Show the solutionHide the solution
- Step 1: The point: y(4) = √4 = 2, so the point is (4, 2).
- Step 2: The slope: y′ = 1/(2√x), so y′(4) = 1/(2·2) = 1/4.
- Step 3: Point-slope form: y − 2 = (1/4)(x − 4).
Answer: y − 2 = (1/4)(x − 4), or y = (1/4)x + 1
- Example 3
Trap: a coefficient in the denominator
Differentiate y = 1/(3x²).
Show the solutionHide the solution
- Step 1: The 3 stays in the bottom as a number: y = (1/3)·x⁻². It is not 3x⁻².
- Step 2: Differentiate: y′ = (1/3)(−2)x⁻³ = −(2/3)x⁻³.
- Step 3: Rewritten: y′ = −2/(3x³).
Answer: y′ = −2/(3x³)
Common mistakes
- Lowering a negative exponent the wrong way: d/dx x⁻³ is −3x⁻⁴, not −3x⁻².
- Applying the power rule to a constant like π² and getting 2π. The derivative of any constant is 0.
- Moving a coefficient along with x when rewriting, such as turning 5/x² into 5⁻¹x⁻² or 5x².
On the exam
- Power rule work shows up inside almost every derivative question. Fast, careful rewriting is worth practicing.
- On free response, an equation of a tangent line can be left in point-slope form; you don't need to simplify.
Connected topics
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Check yourself
4 questions on 2.5 Applying the Power Rule. Pick an answer to see if you got it, and why.
If f(x) = 6√x − 8/x² + 5, what is f′(4)?
If y = 1/∛(x²), then dy/dx =
If f(x) = (x³ − 4x)/√x, what is f′(4) ?
At what point on the graph of y = x^(3/2) is the tangent line parallel to the line y = 3x + 1 ?
0 of 4 answered