# AP Calculus AB Power rule

> AP Calculus AB · Differentiation: Definition and Fundamental Properties
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u2-power-rule/

This module covers the power rule for differentiation, including applications to positive, negative, and fractional exponents, rewriting radicals and reciprocals, and finding tangent lines to power functions, aligned to AP Calculus AB CED requirements.

**Prerequisites:** [Limit definition of the derivative](https://www.owlsprep.com/study/ap-calculus-ab-u2-limit-definition-derivative/); Exponent rules for negative and fractional powers; Point-slope form of a line

## Learning objectives

- Apply the power rule to differentiate any power function
- Handle negative and fractional exponents correctly when differentiating
- Rewrite radicals and reciprocals to use the power rule
- Find the equation of a tangent line to a power function

## Core Power Rule for Positive Integer Exponents

The power rule is the most fundamental differentiation shortcut for AP Calculus AB, allowing you to compute derivatives of power functions without using the limit definition every time. A power function is any function of the form $f(x) = kx^n$, where $k$ is a constant coefficient and $n$ is any real number. This topic makes up 10-12% of Unit 2 exam score.

**Power Rule (with Constant Multiple)** — For any constant $c$ and real number $n$, the derivative of $cx^n$ is found by multiplying the coefficient by the exponent, then subtracting 1 from the exponent. A constant term (where $n=0$) has a derivative of 0.

*Notation:* \frac{d}{dx}[cx^n] = cn x^{n-1}

*Example:* $f(x) = 3x^2$ has derivative $f'(x) = 6x$.

**Worked example:** Find the derivative of $f(x) = 4x^5 - 7x^3 + 12$, then evaluate $f'(1)$.

1. Split the function into individual terms using the sum/difference rule, differentiate each separately:
2. $$f'(x) = \frac{d}{dx}(4x^5) - \frac{d}{dx}(7x^3) + \frac{d}{dx}(12)$$
3. Apply the power rule to the first term: $c=4$, $n=5$, multiply coefficient by exponent and subtract 1 from the exponent:
4. $$4 \cdot 5 x^{5-1} = 20x^4$$
5. Apply the power rule to the second term: $c=7$, $n=3$:
6. $$7 \cdot 3 x^{3-1} = 21x^2$$
7. Apply the power rule to the constant term: $12 = 12x^0$, so derivative is:
8. $$12 \cdot 0 x^{-1} = 0$$
9. Combine terms to get the general derivative, then evaluate at $x=1$:
10. $$f'(x) = 20x^4 - 21x^2 \implies f'(1) = 20(1) - 21(1) = -1$$

> **Exam tip:** Always compute the general derivative first before plugging in the $x$-value for evaluation. Never substitute the $x$-value into the original function before differentiating, as this will always incorrectly give you a slope of zero.

## Power Rule for Negative and Fractional Exponents

The power rule works for any real exponent, not just positive integers. This is one of the most frequently tested aspects of the rule on the AP exam, because it requires you to first rewrite radicals and reciprocals using exponent rules before applying differentiation.

Reciprocals $\frac{1}{x^k}$ rewrite to $x^{-k}$, and $k$-th roots $\sqrt[k]{x}$ rewrite to $x^{1/k}$. After rewriting, you apply the power rule exactly the same way as for positive integers.

**Worked example:** Find the derivative of $g(x) = \frac{6}{x^3} - 4\sqrt[4]{x} + 9x$.

1. First rewrite all terms with standard exponents:
2. $$g(x) = 6x^{-3} - 4x^{1/4} + 9x^1$$
3. Differentiate the first term, with $n=-3$:
4. $$6(-3)x^{-3-1} = -18x^{-4}$$
5. Differentiate the second term, with $n=1/4$:
6. $$-4\left(\frac{1}{4}\right)x^{(1/4 - 1)} = -1x^{-3/4}$$
7. Differentiate the third term, with $n=1$:
8. $$9(1)x^{0} = 9$$
9. Rewrite back to the original radical/reciprocal form for the final answer:
10. $$g'(x) = -\frac{18}{x^4} - \frac{1}{\sqrt[4]{x^3}} + 9$$

> **Exam tip:** Always explicitly write out your exponent subtraction step on paper for negative/fractional exponents. It is extremely common to accidentally subtract 1 from the coefficient instead of the exponent, or get the sign of the new exponent wrong.

## Finding Tangent Lines with the Power Rule

A very common AP exam question asks you to find the equation of a tangent line to a power function curve at a given point. A tangent line to $y=f(x)$ at $x=a$ has two key properties: it passes through the point $(a, f(a))$, and its slope equals the derivative $f'(a)$, which you can compute quickly with the power rule. Once you have slope and a point, use point-slope form to write the final equation.

**Worked example:** Find the equation of the tangent line to $y = 2x^3 - 4x + 3$ at $x=2$.

1. First find the $y$-coordinate of the point by plugging $x=2$ into the original function:
2. $$y = 2(2^3) - 4(2) + 3 = 11, so the tangent passes through (2, 11)$$
3. Compute the derivative using the power rule:
4. $$y' = 2(3)x^2 - 4(1)x^0 + 0 = 6x^2 - 4$$
5. Calculate the slope by evaluating the derivative at $x=2$:
6. $$m = y'(2) = 6(2^2) - 4 = 20$$
7. Substitute into point-slope form, then simplify to slope-intercept form:
8. $$y - 11 = 20(x - 2) \implies y = 20x - 29$$

> **Exam tip:** Never forget to calculate the $y$-coordinate from the original function. Many students only use the derivative to get slope and the given $x$-coordinate, leading to an incorrect $y$-intercept for the tangent line.

## AP-Style Worked Practice Problems

**Worked example:** Multiple Choice: Which of the following is the derivative of $f(x) = \frac{3x^4 - \sqrt{x}}{2x}$?

1. First simplify the original function by dividing each term by $2x$ to write it as a sum of power terms:
2. $$f(x) = \frac{3x^4}{2x} - \frac{\sqrt{x}}{2x} = \frac{3}{2}x^3 - \frac{1}{2}x^{-1/2}$$
3. Apply the power rule to each term: Derivative of $\frac{3}{2}x^3$ is $\frac{3}{2} \cdot 3x^2 = \frac{9}{2}x^2$.
4. Derivative of $-\frac{1}{2}x^{-1/2}$ is $-\frac{1}{2} \cdot (-\frac{1}{2})x^{-3/2} = \frac{1}{4x\sqrt{x}}$.
5. Combine terms for the final derivative, which matches option A:
6. $$f'(x) = \frac{9}{2}x^2 - \frac{1}{4x\sqrt{x}}$$

**Worked example:** Application: The total cost (in dollars) to produce $x$ units of coffee mugs is $C(x) = 500 + 2x + 0.01x^2$, where $500$ is fixed cost. Marginal cost is the derivative of total cost. Find marginal cost at $x=100$ and interpret the result.

1. Differentiate $C(x)$ term by term using the power rule:
2. $$MC(x) = C'(x) = 0 + 2(1)x^0 + 0.01(2)x^{1} = 2 + 0.02x$$
3. Evaluate at $x=100$:
4. $$MC(100) = 2 + 0.02(100) = 4$$
5. Interpretation: After producing 100 mugs, the cost to produce one additional coffee mug is approximately \$4.

## Common pitfalls

- **Wrong:** Differentiating $f(x) = 4x^3$ to get $f'(x) = 4x^2$.
  - Why it fails: Forgets to multiply the original coefficient by the exponent, only subtracts 1 from the exponent.
  - Correct: Always multiply the original coefficient by the exponent first, then subtract 1 from the exponent, giving $f'(x) = 12x^2$.
- **Wrong:** Differentiating $g(x) = x^{-4}$ to get $g'(x) = -4x^{-3}$.
  - Why it fails: Incorrectly subtracts 1 from a negative exponent, resulting in a less negative exponent instead of a more negative one.
  - Correct: Explicitly compute the new exponent on paper: $-4 - 1 = -5$, so the correct derivative is $g'(x) = -4x^{-5}$.
- **Wrong:** Leaving the derivative of $f(x) = 2x^2 + 9$ as $f'(x) = 4x + 9$.
  - Why it fails: Forgets that a constant term has a derivative of zero, since it can be written as $c x^0$.
  - Correct: Mark all constant terms first, write their derivative as 0 before differentiating variable terms, giving $f'(x) = 4x$.
- **Wrong:** Differentiating $h(x) = 3\sqrt{2x}$ to get $h'(x) = \frac{3}{2\sqrt{2x}}$.
  - Why it fails: Forgets that the constant coefficient $\sqrt{2}$ needs to be multiplied into the derivative.
  - Correct: Rewrite as $3\sqrt{2} x^{1/2}$, so derivative is $3\sqrt{2} \cdot \frac{1}{2}x^{-1/2} = \frac{3\sqrt{2}}{2\sqrt{x}}$.
- **Wrong:** Rewriting $\frac{5}{x\sqrt{x}}$ as $5x^{-3/2}$, then differentiating to get $-\frac{5}{3}x^{-5/2}$.
  - Why it fails: Incorrectly multiplies the exponent with the coefficient, flipping the order of operations.
  - Correct: Compute new coefficient first: $5 \cdot (-3/2) = -15/2$, then new exponent: $-3/2 - 1 = -5/2$, so correct derivative is $-\frac{15}{2}x^{-5/2}$.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Basic Power Rule | \frac{d}{dx}[x^n] = n x^{n-1} | Works for any real $n$ (integer, fraction, negative) |
| Power Rule + Constant Multiple | \frac{d}{dx}[cx^n] = cn x^{n-1} | Always multiply original coefficient $c$ by $n$ |
| Derivative of a Constant | \frac{d}{dx}[c] = 0 | Follows from power rule: $c = cx^0$ |
| Reciprocal Term Derivative | \frac{d}{dx}\left[\frac{1}{x^k}\right] = -\frac{k}{x^{k+1}} | Rewrite as $x^{-k}$ before applying rule |
| nth Root Derivative | \frac{d}{dx}\left[\sqrt[k]{x}\right] = \frac{1}{k x^{\frac{k-1}{k}}} | Rewrite as $x^{1/k}$ before applying rule |
| Tangent Line at $x=a$ | y - f(a) = f'(a)(x - a) | $f'(a)$ = slope, $f(a)$ = y-coordinate |
| Derivative of Linear Term | \frac{d}{dx}[mx + b] = m | Derivative of $mx$ is $m$, derivative of $b$ is 0 |

## What's next

The power rule is the foundational building block for every differentiation technique you will learn for the rest of the AP Calculus AB course. Next, you will combine the power rule with the product rule, quotient rule, and eventually the chain rule to differentiate more complex functions, including products of functions, rational functions, and composite functions. Without mastering the power rule—including correctly handling negative and fractional exponents—none of these more advanced techniques will be possible, as every differentiation ultimately reduces to applying the power rule to individual terms. Beyond differentiation, the power rule is also critical for integration, where you reverse the power rule to find antiderivatives for polynomial and power functions, a core skill for the integral units later in the course.

- [The chain rule](https://www.owlsprep.com/study/ap-calculus-ab-u3-chain-rule/)
- [Constant, sum, difference, and constant multiple rules](https://www.owlsprep.com/study/ap-calculus-ab-u2-constant-sum-difference-and-constant/)
- [Derivatives of cos, sin, e^x, ln(x)](https://www.owlsprep.com/study/ap-calculus-ab-u2-derivatives-of-cos-sin-e/)

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