Calculus I · Unit 2A · lesson

Negative and Fractional Powers

Concept

Learning objectives

Apply the power rule to radicals and reciprocal powers by rewriting them with exponents.

Rewrite Before Differentiating

Explanation

Before the formulas

The shortcuts in Negative and Fractional Powers do not replace the limit definition. They package conclusions already justified from it. That distinction matters because a rule applies only when the function has the relevant structure and lies in its domain. A familiar-looking exponent or fraction is not permission to differentiate blindly.

After using a rule, check the result structurally. A derivative of a polynomial should have lower degree. A product derivative should usually contain contributions from both factors. A quotient result should preserve the denominator restriction. These checks are quick enough to use on homework and valuable enough to use on exams.

Explanation

Rewrite before differentiating

Expressions such as $1/x^3$ and $\sqrt[4]{x^7}$ are power functions in disguise. Writing them as $x^{-3}$ and $x^{7/4}$ lets the power rule operate without inventing separate rules for every radical and reciprocal.

The rewritten form also exposes domain issues. A derivative formula may be algebraically correct but defined only where the original function and the relevant powers make sense. Calculus answers live on domains, not in a symbol vacuum.

Radicals and reciprocals are power functions wearing different notation. Rewriting them with rational or negative exponents reveals the structure the power rule needs. This is not cosmetic algebra; it is a way of making the function's architecture visible.

Domains matter more here. A symbolic derivative formula may exist on intervals where the original real-valued function is defined, but not beyond them. For example, $\sqrt{x}$ and its derivative are real only on appropriate domains, and the derivative becomes unbounded as $x\to0^+$ .

The power rule is easiest to use when every variable factor is written as $x^n$ :

\frac1{x^m}=x^{-m},\qquad \sqrt{x}=x^{1/2},\qquad \sqrt[3]{x^2}=x^{2/3}.

Guided walkthrough

A reciprocal and a square root

Differentiate

f(x)=\frac{4}{x^3}-5\sqrt{x}.

Answer reveal

Worked solution

Your setup or answer

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Worked example

A cube-root power

For $g(x)=\sqrt[3]{x^5}=x^{5/3}$ ,

g'(x)=\frac53x^{2/3}.

This derivative exists for every real $x$ , including $0$ .

Exam note

Do not rewrite $1/(x^2+1)$ as $x^{-2}+1$ . A negative exponent applies to the entire base inside its parentheses. The function $(x^2+1)^{-1}$ requires the chain rule, introduced later.

Exercise

Differentiate $x^{-7}$ .

Exercise

Differentiate $6/\sqrt{x}$ .

Exercise

Differentiate $3\sqrt[4]{x^3}$ .

Exercise

Find the tangent slope of $f(x)=\sqrt{x}$ at $x=9$ .

Exercise

State the domain on which the derivative of $x^{2/3}$ is finite.

Application

Stopping distance and square-root speed

In a simplified braking model, the maximum safe speed on a fixed stopping distance $d$ may have the form

v(d)=k\sqrt d.

Then

v'(d)=\frac{k}{2\sqrt d}.

Increasing available distance helps most when the original stopping distance is small; the marginal gain decreases as $d$ grows. The fractional exponent makes that diminishing sensitivity visible.

After the explanation

Use the section idea

Reading lens

Every rule is a compressed limit calculation; choose the structure before doing algebra.

Mental model

Sums contribute independently, products have two changing contributions, and quotients must account for a changing denominator.

Decision

Name the outermost algebraic structure, then apply the smallest rule set that preserves it.

Common trap

Applying a familiar rule to the wrong outer structure or simplifying after a differentiation error.

Check yourself

Can you justify the primary rule before writing the first derivative symbol?

Interactive checkpower-extra-02

Differentiate $x^{-3}$ .

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Show hint

Use the power rule with exponent $-3$ .

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