AP Calculus AB · Lesson review · September 9, 2026

The Power Rule and Basic Derivative Rules

See why the power rule works, then use it confidently on positive, negative, and fractional powers.

15-minute edited lesson12 chaptersCaptions + corrected transcript

01 · Watch

Lesson video

0:0001The Power Rule 0:1502The Binomial Theorem 1:5803Combinations and Factorials 3:4004State the Power Rule 4:1305Prove the Power Rule 8:0606Positive Integer Powers 8:5407Negative Integer Powers 9:4908Fractional Powers 10:4709Coefficients Need Another Rule 11:4510The Constant-Multiple Rule 11:5911Prove the Constant-Multiple Rule 14:0012The Sum and Difference Rule

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02 · Understand

Learning targets

Read the binomial theorem and connect its coefficients to combinations.
Use the derivative definition and a binomial expansion to justify the power rule for whole-number exponents.
Differentiate positive, negative, and fractional powers.
Keep a constant coefficient while differentiating its function.
Differentiate sums and differences term by term.

03 · Rules

The derivative toolkit

These rules turn the limit definition into efficient differentiation.

01Binomial theorem: $(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$ It expands a whole-number power of a sum. In the proof, every term after the first contains a factor of $h$ .
02Combinations: $\binom{n}{r}=\frac{n!}{r!(n-r)!}$ The coefficients are the entries in the corresponding row of Pascal's triangle.
03Factorial: $n!=n(n-1)(n-2)\cdots2\cdot1$ Factorials make the combination formula compact and reveal common factors that cancel.
04Power rule: $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$ Multiply by the old exponent; the new exponent is one less.
05Constant-multiple rule: $\frac{d}{dx}\left[c\,g(x)\right]=c\,g'(x)$ A constant factor stays in front while the function is differentiated.
06Sum and difference rule: $\frac{d}{dx}\left[f(x)\pm g(x)\right]=f'(x)\pm g'(x)$ Differentiate each term separately and preserve the plus or minus sign.

04 · Apply

Proofs and examples

Rewrite radicals and reciprocals as powers before applying the shortcut.

Why it works

Prove the power rule

\lim_{h\to0}\frac{(x+h)^n-x^n}{h}=nx^{n-1}

Expand $(x+h)^n$ with the binomial theorem.
The leading $x^n$ cancels, and every remaining term contains $h$ .
Cancel the common factor of $h$ with the denominator.
As $h\to0$ , every remaining term containing $h$ approaches zero.

The only surviving term is $nx^{n-1}$ .

Positive exponent

Drop the power

\frac{d}{dx}\left(x^{20}\right)=20x^{19}

The coefficient becomes 20, and the exponent decreases from 20 to 19.

Negative exponent

A reciprocal is still a power

\frac{d}{dx}\left(x^{-4}\right)=-4x^{-5}=-\frac{4}{x^5}

Multiply by −4, subtract one from the exponent, then rewrite with a positive exponent if desired.

Fractional exponent

A radical is still a power

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

Rewrite the square root as an exponent of one-half. The negative exponent in the derivative becomes a radical in the denominator.

Constant coefficient

Keep the constant factor

\frac{d}{dx}\left(5x^3\right)=15x^2

The constant-multiple rule is justified by factoring the constant out of the difference quotient:

\lim_{h\to0}\frac{cg(x+h)-cg(x)}{h}=c\,g'(x)

05 · Avoid

Common mistakes

Forgetting the coefficient.

The old exponent becomes a multiplier.

Leaving the exponent unchanged.

Subtract one after bringing the exponent down.

Treating a radical as a separate rule.

Rewrite it as a fractional power first.

Losing a negative sign.

For $x^{-4}$ , the multiplier is negative four.

Differentiating a constant coefficient.

Keep the constant and differentiate the function it multiplies.

Applying the proof too broadly.

The displayed binomial proof establishes the rule for whole-number powers; the rule extends further with additional results.

06 · Check yourself

Quick check

Try each before revealing the answer.

1Differentiate

x^7

$7x^6$ .

2Differentiate

x^{-3}

and rewrite without a negative exponent.

$-3x^{-4}=-\frac{3}{x^4}$ .

3Differentiate

\sqrt{x}

$\frac{1}{2\sqrt{x}}$ .

4Differentiate

8x^5

$40x^4$ . Keep the 8 and multiply it by the exponent 5.

5Why do all but one term vanish at the end of the power-rule proof?

After cancellation, those terms still contain positive powers of $h$ , and $h\to0$ .

07 · Revisit

Transcript and practice

Search the corrected transcript for a rule or example. Captions follow the edited video timeline.

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Class practice: use the Bluebook Section 2.2 assignment linked in the Wednesday 9/9 eKadence activity.