Polynomial power rule
Rewrite as explicit polynomial terms, differentiate each term, and preserve the original domain
\[(ax^n)\prime=nax^{n-1}\quad(n\ge1),\qquad(c)\prime=0\]
- Differentiate each power For a positive integer power, multiply the coefficient by that power and reduce the power by one \(\displaystyle \frac{d}{dx}(ax^n)=nax^{n-1},\quad n\ge1\)
- A constant has zero derivative A constant does not change as the input changes. Keep a zero entry when the response asks for the constant coefficient \(\displaystyle \frac{d}{dx}(c)=0\)
- Keep addition and scales Differentiate a sum term by term. A fixed rational scale multiplies the derivative too \(\displaystyle \frac{d}{dx}(af(x)+bg(x))=af\prime(x)+bg\prime(x)\)
- Do the algebra first Expand products and scaled brackets, or cancel common factors under the original input restriction. Then apply the polynomial power rule
- Keep the original domain Cancelling an input factor does not define the original function at an excluded input
Worked example
Differentiate the displayed product by first expanding
\[f(x)=(x-1)(2x^2+3)\]
- \(\displaystyle f(x)=2x^3-2x^2+3x-3\)Distribute both terms of the linear factor
- \(\displaystyle f\prime(x)=6x^2-4x+3\)Multiply by each old power, reduce it by one, and remove the constant
- \(\displaystyle (A_2,A_1,A_0)=(6,-4,3)\)The full descending coefficient vector gives one entry for each power through zero
Common mistakes
Changing only the power
✕\(\displaystyle (3x^4)\prime=3x^3\)✓\(\displaystyle (3x^4)\prime=12x^3\)
The original power also multiplies the coefficientKeeping the original constant
✕\(\displaystyle (x^2+7)\prime=2x+7\)✓\(\displaystyle (x^2+7)\prime=2x\)
The constant has no rate of changeRestoring an excluded input
✕After cancelling x, the original rule is defined at zero✓Keep the original restriction excluding zero after cancellation
An algebraically simplified expression agrees only on the original domainDifficulty levels
introDifferentiate constants, monomials and short polynomials
standardUse exact fractional coefficients and explicit missing-power entries
challengeExpand factorised or scaled rules and simplify quotients while preserving their domains
examChoose a derivative rule and produce its full coefficient vector in varied notation
Questions about this topic
Why enter zero for an absent term?+
The labelled basis has a fixed slot for every requested power. Zero states that the derivative has no such term
Do these products require a product rule?+
No. Expand to a polynomial first; the power rule is sufficient for these constructions
Does the coefficient table assess written method?+
It checks exact coefficients. Complete written reasoning and independent mastery require separate evidence
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 6: Applications of differentiation
In this chapter
Differentiate polynomialsChallenge enrichment: rational powers and derivative domainsDifferentiate natural exponential functions and solve simple derivative conditionsDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsDifferentiate a composite function by the chain ruleDifferentiate a product or quotient of two functionsEvaluate and determine function rules6B Differentiation rules: Find the maximal domain of an algebraic functionExpand and factor polynomial expressionsSimplify and combine algebraic fractions6B Differentiation rules: Find the equation of a line from its graph or from its intercept and slopeAnalyse polynomial curves, stationary points and extremaUse a derivative to analyse and recognise the graph of a function6C Tangents, normals and curve sketching: Evaluate a polynomial or recover a coefficient parameter6C Tangents, normals and curve sketching: Move between roots, factors, polynomial rules and equationsRecognise and classify a polynomialConnect a factorised higher-degree rule and its graphAnalyse a cubic graph or determine its ruleWork with sets and interval notation6C Tangents, normals and curve sketching: Find the maximal domain of an algebraic function6C Tangents, normals and curve sketching: Find the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a maximum or minimum problem by differentiationSet up and solve a practical linear equationRearrange literal equations and recognise equivalent line forms6D Optimisation: Find the maximal domain of an algebraic function6D Optimisation: Evaluate a polynomial or recover a coefficient parameter6D Optimisation: Move between roots, factors, polynomial rules and equationsCalculate and interpret average and instantaneous rates from a modelVolume of cones, pyramids and spheresVolume of prisms and cylindersPerimeter and area of standard shapesSimilar figures and scale factorsFind a side of a right-angled triangleAnalyse straight-line motion through displacement, velocity and acceleration6F Kinematics: Evaluate a polynomial or recover a coefficient parameter