Does the power rule work for negative and fractional exponents?

Yes. The rule d/dx[x^n]=nx^n-1 holds for any real exponent, including negatives like x^-3 and fractions like x^1/2. Just rewrite roots and reciprocals as powers of x first, then apply the rule normally.

Why does the constant term in a polynomial disappear when I differentiate?

A constant contributes a horizontal piece to the graph, which has zero slope. Since the derivative measures slope, any standalone constant differentiates to 0 and drops out of the answer.

Can I use linearity on products like x^2· x^3?

Not directly — linearity only covers sums, differences, and constant multiples. For a product you either simplify first (here x^2· x^3=x^5, so the derivative is 5x^4) or use the product rule taught in a later lesson. Never assume the derivative of a product is the product of derivatives.

How do I write derivative answers the way the AP exam expects?

Either form is usually accepted, but it's safest to convert negative and fractional exponents back into fractions and radicals, such as writing 5/2x^-1/2 as 52sqrt(x). Match the form the question uses and keep signs on negative exponents accurate.

AP-CALCBC-2.5-2.6

U2.5 The Power Rule and Linearity of the Derivative

Master the power rule d/dx[xⁿ]=n·x⁽ⁿ⁻¹⁾ for any real n plus the linearity rules to differentiate polynomials and power functions fast on the AP exam.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U2.5 The Power Rule and Linearity of the Derivative, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start U2.5 The Power Rule and Linearity of the Derivative free

What this lesson covers

By now you know a derivative measures instantaneous rate of change, and you've computed a few from the limit definition. That definition is powerful but slow. This lesson gives you the tools that make differentiation quick and reliable: the power rule and the linearity properties. Together they let you differentiate any polynomial, radical, or negative-exponent power of

x

in seconds — no limits required.

These two ideas are the workhorses of the entire course. Almost every AP problem that involves a derivative starts by applying them, so getting them automatic now pays off through related rates, optimization, and beyond. We'll cover the rules, the rewriting tricks that unlock them, and the traps that cost students easy points.

The Power Rule for Any Real Exponent

The power rule states that for any real number

n

\frac{d}{dx}\left[x^{n}\right] = n\,x^{n-1}.

You multiply by the old exponent, then subtract one from the exponent. It works not just for positive integers but for negatives, fractions, and irrational exponents.

Function	Rewrite	Derivative
$x^{5}$	$x^{5}$	$5x^{4}$
$\sqrt{x}$	$x^{1/2}$	$\tfrac{1}{2}x^{-1/2}$
$\dfrac{1}{x^{3}}$	$x^{-3}$	$-3x^{-4}$
$x^{\pi}$	$x^{\pi}$	$\pi x^{\pi-1}$

The key skill is rewriting before differentiating. Roots must become fractional exponents and reciprocals must become negative exponents so the rule applies cleanly. A common misconception is trying to keep

\frac{1}{x^3}

in fraction form and guessing — always convert to

x^{-3}

first.

Two special cases follow directly. The derivative of

x^{1}=x

1x^{0}=1

, matching the slope of the line

y=x

. And any constant like

x^{0}=1

has derivative

0x^{-1}=0

, which agrees with the constant rule below. On the AP exam, expect questions that deliberately disguise a power of

x

as a radical or fraction to test whether you rewrite correctly.

Linearity: Constant, Constant-Multiple, and Sum/Difference

Differentiation is a linear operation, which means three convenient rules hold. If

c

is a constant and

f

g

are differentiable functions:

Rule	Statement
Constant	$\dfrac{d}{dx}[c]=0$
Constant multiple	$\dfrac{d}{dx}[c\,f(x)]=c\,f'(x)$
Sum/difference	$\dfrac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$

The constant rule reflects that a horizontal line has slope zero. The constant-multiple rule lets you pull numbers outside the derivative: to differentiate

7x^{4}

, differentiate

x^{4}

to get

4x^{3}

, then multiply by 7 for

28x^{3}

.

The sum/difference rule means you can differentiate a polynomial term by term. There is no special "rule" needed for adding functions — you simply handle each piece separately and keep the plus or minus signs.

A frequent error is forgetting that a lone constant term vanishes. In

f(x)=3x^{2}+9

, the

9

contributes nothing to

f'(x)

. Another is mishandling a coefficient in a denominator, like

\frac{x^2}{4}

; treat it as

\frac{1}{4}x^2

so the constant multiple is clearly

\frac{1}{4}

. Importantly, linearity does NOT extend to products or quotients —

\frac{d}{dx}[f\cdot g]\neq f'\cdot g'

. Those need the product and quotient rules in later lessons.

Combining the Rules to Differentiate Polynomials

Almost every early derivative problem combines the power rule with linearity. The strategy is always the same: rewrite each term as a power of

x

, apply the power rule term by term, then multiply by any coefficients and drop constants.

Consider

f(x)=4x^{3}-\dfrac{2}{x}+5\sqrt{x}-7

. Rewrite it as

f(x)=4x^{3}-2x^{-1}+5x^{1/2}-7.

Differentiate each term:

f'(x)=12x^{2}+2x^{-2}+\tfrac{5}{2}x^{-1/2}.

Notice the middle term:

-2x^{-1}

differentiates to

-2(-1)x^{-2}=+2x^{-2}

, and the sign flips because the exponent was negative. The constant

-7

disappears.

The AP exam tests this constantly, both on the multiple-choice (find

f'

or evaluate

f'(a)

) and inside larger free-response problems where a derivative is just step one. A clean answer is often expected back in radical or fraction form, so you may need to rewrite

\frac{5}{2}x^{-1/2}

\frac{5}{2\sqrt{x}}

. Always double-check signs on negative exponents and remember to distribute coefficients before applying the exponent decrease.

Using Derivatives to Find Slopes and Tangent Lines

Once you have

f'(x)

, evaluating it at a point gives the slope of the tangent line there. This connects the mechanical rules to the geometric meaning of a derivative from earlier lessons.

To find the tangent line to

y=f(x)

x=a

:

First compute

f'(x)

using the power rule and linearity. Then evaluate

m=f'(a)

for the slope, and

f(a)

for the point. Finally write the point-slope form

y-f(a)=m(x-a)

.

The same derivative answers other AP favorites: where is the tangent horizontal (solve

f'(x)=0

), and what is the instantaneous rate of change at a specific input. A horizontal tangent occurs precisely where the derivative equals zero, since a slope of zero is a flat line.

A subtle point tested often: the question may give you the derivative value and ask you to work backward, or provide a graph of

f

and ask about the sign of

f'

. Keep straight that

f'(a)

is a number (the slope at one point), while

f'(x)

is a whole function. Confusing the two — for instance, plugging in before differentiating — is a classic mistake. Differentiate first, substitute second.

Key terms

Power Rule.: The rule $\frac{d}{dx}[x^{n}]=n\,x^{n-1}$ , valid for any real exponent $n$ .
Linearity of the Derivative.: The property that the derivative of a sum is the sum of derivatives and constants factor out: $\frac{d}{dx}[c\,f\pm g]=c\,f'\pm g'$ .
Constant Rule.: The derivative of any constant is zero, since a horizontal line has slope zero.
Constant-Multiple Rule.: A constant coefficient can be pulled outside the derivative: $\frac{d}{dx}[c\,f(x)]=c\,f'(x)$ .
Fractional Exponent.: A root written as a power, such as $\sqrt{x}=x^{1/2}$ , so the power rule can be applied.
Negative Exponent.: A reciprocal written as a power, such as $\frac{1}{x^{n}}=x^{-n}$ , enabling the power rule.
Tangent Line.: The line touching a curve at a point with slope equal to the derivative there, given by $y-f(a)=f'(a)(x-a)$ .

Worked example

Let

f(x)=2x^{4}-\dfrac{3}{x^{2}}+6\sqrt[3]{x}-11

. Find

f'(x)

, then find the slope of the tangent line to

f

x=1

First rewrite every term as a power of

x

. The reciprocal

\frac{3}{x^{2}}

becomes

3x^{-2}

, and the cube root

\sqrt[3]{x}

becomes

x^{1/3}

f(x)=2x^{4}-3x^{-2}+6x^{1/3}-11.

Now differentiate term by term. For

2x^{4}

2\cdot 4x^{3}=8x^{3}

. For

-3x^{-2}

-3\cdot(-2)x^{-3}=6x^{-3}

. For

6x^{1/3}

6\cdot\tfrac{1}{3}x^{-2/3}=2x^{-2/3}

. The constant

-11

gives

0

. So

f'(x)=8x^{3}+6x^{-3}+2x^{-2/3}.

Written without negative exponents,

f'(x)=8x^{3}+\dfrac{6}{x^{3}}+\dfrac{2}{x^{2/3}}

.

Now evaluate at

x=1

. Since

1

raised to any power is

1

f'(1)=8(1)+6(1)+2(1)=16

.

The slope of the tangent line at

x=1

16

. Notice the exponent on the negative-exponent term became more negative, and its sign flipped from negative to positive during differentiation — a detail worth double-checking.

Practice questions

f(x)=\dfrac{5}{x^{2}}+3x

, what is

f'(x)

$-\dfrac{10}{x^{3}}+3$
$\dfrac{10}{x^{3}}+3$
$-\dfrac{5}{x^{3}}+3$
$-\dfrac{10}{x^{3}}+3x$

Answer: $-\dfrac{10}{x^{3}}+3$

Rewrite as

f(x)=5x^{-2}+3x

. The power rule gives

5\cdot(-2)x^{-3}=-10x^{-3}=-\frac{10}{x^{3}}

, and

3x

differentiates to

3

. Constants pulled through correctly yield

-\frac{10}{x^{3}}+3

. The trap answer keeps a positive sign or an

x

where a constant should appear.

Find all

x

-values where the tangent line to

g(x)=x^{3}-12x+4

is horizontal.

Answer: $x=2$ and $x=-2$

A horizontal tangent occurs where

g'(x)=0

. Differentiating term by term gives

g'(x)=3x^{2}-12

. Setting

3x^{2}-12=0

gives

x^{2}=4

, so

x=2

x=-2

. Both are valid because the derivative is a function defined for all real

x

; you must report both solutions.

Explain why the derivative of

f(x)=7

0

, and connect this to the power rule.

Answer: The graph is a horizontal line with slope 0, and treating $7$ as $7x^{0}$ gives $7\cdot 0\cdot x^{-1}=0$ .

Geometrically, a constant function never changes, so its instantaneous rate of change is zero everywhere. Algebraically, write

7=7x^{0}

; applying the constant-multiple and power rules gives

7\cdot(0\cdot x^{-1})=0

. This shows the constant rule is consistent with the power rule rather than a separate idea.

FAQ

Does the power rule work for negative and fractional exponents?: Yes. The rule $\frac{d}{dx}[x^n]=nx^{n-1}$ holds for any real exponent, including negatives like $x^{-3}$ and fractions like $x^{1/2}$ . Just rewrite roots and reciprocals as powers of $x$ first, then apply the rule normally.
Why does the constant term in a polynomial disappear when I differentiate?: A constant contributes a horizontal piece to the graph, which has zero slope. Since the derivative measures slope, any standalone constant differentiates to $0$ and drops out of the answer.
Can I use linearity on products like $x^2\cdot x^3$ ?: Not directly — linearity only covers sums, differences, and constant multiples. For a product you either simplify first (here $x^2\cdot x^3=x^5$ , so the derivative is $5x^4$ ) or use the product rule taught in a later lesson. Never assume the derivative of a product is the product of derivatives.
How do I write derivative answers the way the AP exam expects?: Either form is usually accepted, but it's safest to convert negative and fractional exponents back into fractions and radicals, such as writing $\frac{5}{2}x^{-1/2}$ as $\frac{5}{2\sqrt{x}}$ . Match the form the question uses and keep signs on negative exponents accurate.

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