Calculus I · 2A · lesson
Negative and Fractional Powers
Learn negative and fractional powers with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Core differentiation rulesWhat this section is building
Learn negative and fractional powers with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Sums contribute independently, products have two changing contributions, and quotients must account for a changing denominator.
Name the outermost algebraic structure, then apply the smallest rule set that preserves it.
Applying a familiar rule to the wrong outer structure or simplifying after a differentiation error.
Learning objectives
Apply the power rule to radicals and reciprocal powers by rewriting them with exponents.
Rewrite Before Differentiating
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.
Rewrite before differentiating
Expressions such as and are power functions in disguise. Writing them as and 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, and its derivative are real only on appropriate domains, and the derivative becomes unbounded as .
The power rule is easiest to use when every variable factor is written as :
A reciprocal and a square root
Differentiate
Worked solution
Write a real attempt before opening the supplied answer.
A cube-root power
For ,
This derivative exists for every real , including .
Do not rewrite as . A negative exponent applies to the entire base inside its parentheses. The function requires the chain rule, introduced later.
Differentiate .
Differentiate .
Differentiate .
Find the tangent slope of at .
State the domain on which the derivative of is finite.
Stopping distance and square-root speed
In a simplified braking model, the maximum safe speed on a fixed stopping distance may have the form
Then
Increasing available distance helps most when the original stopping distance is small; the marginal gain decreases as grows. The fractional exponent makes that diminishing sensitivity visible.
power-extra-02Differentiate .
Your work stays on this device. No account or AI grader is used.
Show hint
Use the power rule with exponent .
Attempt once to unlock the solution
Submit an answer first. The hint is available now.
Source & rights
Original instruction with traceable references.
BetterGrades-original composition declared by source handoff; owner provenance review required before public release
Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.