Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Differentiate constants and linear combinations, preserving signs before point evaluation.
College Board Topic 2.6 · CED p. 60 (PDF p. 65) · CED effective Fall 2020; project course outline 2026–27.
A constant function has derivative 0 because its output does not change. Derivatives pass through sums, differences and constant multiples: (af+bg)′=af′+bg′ when the component derivatives exist. The coefficients keep their signs.
Combine those rules with the power rule to differentiate a polynomial term by term. For P(x)=3x⁴−2x²+7x−9, P′(x)=12x³−4x+7. The final constant contributes 0, while the linear term contributes its slope 7.
A derivative is still a function, so evaluate it at the requested input only after differentiating. Replacing x by a number too early turns the expression into a constant and loses its variation. These linearity rules do not say that the derivative of a product is the product of derivatives.
Let's work through it
- Differentiate 3x⁴ using the constant multiple and power rules to get 12x³. Reason: the coefficient stays while the exponent moves and decreases.
- Differentiate −2x², 7x and −9 to get −4x, 7 and 0. Reason: sums differentiate term by term and constants have zero rate.
- Combine to get P′(x)=12x³−4x+7. Reason: the sum and difference rules preserve all signs.
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 4x³−5x+8.
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Differentiate 4x³−5x+8.
Worked answer and reason
12x²−5.
Use linearity, the power rule and the constant rule.
Retrieval 2
Why should you not set x=2 before finding f′(2)?
Worked answer and reason
That would turn f(x) into a single constant value.
Differentiate the function first, then evaluate its slope.
Catch the tempting mistake
- The derivative of −5x is 0 because −5 is constant.
- The entire term varies with x; its derivative is the coefficient −5.
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 17 · No calculator · analytic
- Unit 2 problem 18 · No calculator · analytic
For the teacher
- Use sign-sensitive polynomials as a short retrieval routine.
- Ask students which terms describe constant slopes and which describe varying slopes.
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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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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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.
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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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Polynomial
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable. The degree is the highest exponent with a nonzero coefficient. For example, 2x³−x+5 is a polynomial of degree 3. Expressions containing 1/x or √x are not polynomials in x.