The power rule is a shortcut for finding the derivative of any function shaped like $f(x)=x^r$ . The rule says $f'(x)=r \cdot x^{r-1}$ , so you bring the exponent down in front and subtract one from it. For AP Calculus, rewrite roots and denominator powers as exponents first so the rule applies cleanly.

Why This Matters for the AP Calculus Exam

The power rule is one of the first derivative shortcuts in AP Calculus, and you will use it constantly for the rest of the course. Almost every derivative you take later, including those that need the product rule, quotient rule, or chain rule, depends on quickly differentiating power functions. On multiple-choice and free-response questions, you are expected to calculate derivatives of familiar functions like $x^r$ accurately and without the full limit definition. Building fluency here makes polynomial derivatives, tangent line slopes, and later applications much faster.

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Key Takeaways

For $f(x) = x^r$ , the derivative is $f'(x) = r \cdot x^{r-1}$ .
The rule works for any real exponent $r$ , including negative and fractional powers.
Rewrite roots and fractions as exponents before differentiating, for example $\sqrt{x} = x^{1/2}$ and $\frac{1}{x^5} = x^{-5}$ .
You can confirm the power rule with the limit definition of the derivative, but the shortcut saves time.
The derivative of a constant is $0$ , which matters when a power function is part of a larger expression.

The Power Rule

The power rule states that if $f(x) = x^r$ , where $r$ is a constant, then the derivative $f'(x)$ is:

$f'(x) = r \cdot x^{r-1}$

So you bring the exponent down as a coefficient, then reduce the exponent by 1. This gives you a fast way to find the derivative without using the limit definition of a derivative. Both methods give the same answer, but the power rule is much quicker.

The rule applies to more than just whole number exponents. Negative exponents, like $x^{-5}$ , and fractional exponents, like $x^{1/2}$ , all follow the same pattern. The trick is to rewrite the function in $x^r$ form first.

A few quick examples:

$\frac{d}{dx}\sqrt{x} = \frac{d}{dx} x^{1/2} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$
$\frac{d}{dx} x^{-2} = -2x^{-3} = \frac{-2}{x^3}$

How to Use This on the AP Calculus Exam

Problem Solving

Before you apply the power rule, get every term into the form $x^r$ . Roots become fractional exponents and fractions with $x$ in the denominator become negative exponents. Skipping this rewrite is the most common reason students miss these.

Common Trap

Watch your arithmetic when the exponent is negative or fractional. Subtracting 1 from $-5$ gives $-6$ , not $-4$ , and subtracting 1 from $\frac{1}{2}$ gives $-\frac{1}{2}$ . Keeping clear notation helps you avoid sign and fraction errors, which is important for clear exam work.

Practice Problems

Try these, then check your work below.

Given $f(x) = x^4$ , find $f'(x)$ .
Given $f(x) = \frac{1}{x^5}$ , find $f'(x)$ .
Given $f(x) = \sqrt{x}$ , find $f'(x)$ .
Given $f(x) = x^6 + 2x^4 - 10$ , find $f'(x)$ .

Before you check, remember:

The power rule with fractions can be tricky, so rewriting the function helps.
The derivative of any constant is zero.

Answers

Notice how several problems are rewritten before differentiating.

$f'(x) = 4 \cdot x^{4-1} = 4x^3$
$f(x) = x^{-5}$ , so $f'(x) = -5 \cdot x^{-5-1} = -5x^{-6} = \frac{-5}{x^6}$
$f(x) = x^{1/2}$ , so $f'(x) = \frac{1}{2} \cdot x^{-1/2} = \frac{1}{2\sqrt{x}}$
$f'(x) = 6x^5 + 8x^3$

For problem 4, each power term follows the power rule and the constant $-10$ has a derivative of $0$ .

Once you are comfortable here, you can move on to the rest of the derivative rules you need to know.

Common Misconceptions

The power rule only works for $x$ raised to a constant power. It does not directly apply to expressions like $2^x$ , where the base is constant and the exponent is the variable.
You must rewrite roots and fractions first. The rule does not apply to $\sqrt{x}$ or $\frac{1}{x^5}$ until you write them as $x^{1/2}$ and $x^{-5}$ .
Bringing the exponent down does not mean you drop it from the variable. You reduce the exponent by 1, you do not erase it.
A negative exponent stays negative when you subtract 1. For $x^{-5}$ , the new exponent is $-6$ , which makes the term smaller, not larger.
The derivative of a constant term is $0$ , not the constant itself.

Vocabulary

The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.

Term	Definition
definition of the derivative	The formal mathematical definition using limits: f'(x) = lim(h→0) [f(x+h) - f(x)]/h, which defines the derivative as the instantaneous rate of change.
derivative	The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point.
power rule	A derivative rule stating that the derivative of x^n is n·x^(n-1), where n is a constant.

Frequently Asked Questions

What is the power rule in calculus?

The power rule says that if f(x) = x^r, then f'(x) = r x^(r - 1). Bring the exponent down, then subtract 1 from the exponent.

What is the derivative of x?

The derivative of x is 1 because x = x^1, so the power rule gives 1 times x^0, which equals 1.

What is the derivative of 5x^4?

Using the constant multiple rule and power rule, the derivative of 5x^4 is 20x^3.

How do you find the derivative of square root of x?

Rewrite square root of x as x^(1/2), then apply the power rule: the derivative is (1/2)x^(-1/2), or 1/(2 square root x).

Does the power rule work with negative exponents?

Yes. Rewrite fractions with x in the denominator as negative powers, then subtract 1 from the exponent. For example, x^-5 becomes -5x^-6.

When should you not use the power rule directly?

Do not use the power rule directly when the variable is in the exponent, like 2^x, or when the expression is not rewritten as x raised to a constant power.