Applying the Power Rule
Apply the power rule to integer, negative, and fractional powers on the valid real domain.
College Board Topic 2.5 · CED p. 59 (PDF p. 64) · CED effective Fall 2020; project course outline 2026–27.
For f(x)=xʳ, the power rule gives f′(x)=r x^(r−1) wherever that real function and derivative are defined. The exponent moves to the front, then decreases by 1. The rule follows from the derivative definition; it is a shortcut for the limiting quotient, not a replacement for the concept.
For x² the quotient becomes 2x+h and tends to 2x. For x³ expansion gives 3x²+3xh+h² and tends to 3x². The general rule also handles negative and fractional powers: (x^(−2))′=−2x^(−3) for x≠0, and (√x)′=1/(2√x) for x>0.
Domain and endpoint behavior matter. √x is defined and right-continuous at 0 but has no finite derivative there; x^(1/3) is defined for negative inputs but its derivative formula excludes 0. Rewrite radicals and reciprocals as powers before differentiating, then state the valid domain.
Let's work through it
- Rewrite 1/x³ as x^(−3), x≠0. Reason: power notation exposes the exponent.
- Apply the power rule: −3x^(−4). Reason: multiply by −3 and reduce the exponent by 1.
- Rewrite as −3/x⁴ and retain x≠0. Reason: algebraic simplification does not expand the original domain.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Differentiate x^(5/2) for x>0.
See every recall question with its answer
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Retrieval 1
Differentiate x^(5/2) for x>0.
Worked answer and reason
(5/2)x^(3/2).
The exponent decreases by 1.
Retrieval 2
Where is the derivative formula for √x valid?
Worked answer and reason
For x>0.
The denominator 2√x is zero at 0; no finite endpoint slope exists there.
Catch the tempting mistake
- The power rule adds 1 to the exponent.
- Differentiation subtracts 1; adding 1 belongs to a later antiderivative rule.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 2 problem 14 · No calculator · analytic
- Unit 2 problem 15 · No calculator · analytic
- Unit 2 problem 16 · No calculator · analytic
For the teacher
- Mix integer, negative and rational exponents, with explicit real domains.
- Use the definition once to show what the shortcut is computing.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Power rule
The derivative power rule gives d(xⁿ)/dx=n xⁿ⁻¹ wherever the real power function is differentiable. Constant multiples remain as constant factors. For example, the derivative of 3x⁴ is 12x³. A composite power such as (x²+1)³ also requires the chain rule.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.