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Differentiate power functions and evaluate a derivative

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The power rule

Rewrite each term as a constant multiplied by a power of x. Multiply the coefficient by the exponent, reduce the exponent by one and differentiate constants to zero

\[\frac{d}{dx}\left(ax^{n}\right)=anx^{n-1}\]
  1. 1Rewrite roots and reciprocals Write roots as fractional powers and reciprocals as negative powers before applying the rule \(\displaystyle \sqrt{x}=x^{1/2},\quad \frac{1}{x^{2}}=x^{-2}\)
  2. 2Multiply the coefficient and exponent For a term ax^n, the new coefficient is a multiplied by n. \(\displaystyle 5x^{3}\quad\Rightarrow\quad 5\times3x^{2}=15x^{2}\)
  3. 3Reduce the exponent by one After multiplying the coefficient, replace n by n-1 in the power \(\displaystyle x^{n}\quad\Rightarrow\quad nx^{n-1}\)
  4. 4Drop constant terms A constant has no change as x changes, so its derivative is zero \(\displaystyle \frac{d}{dx}\left(-7\right)=0\)
Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

The power rule

Rewrite each term as a constant multiplied by a power of x. Multiply the coefficient by the exponent, reduce the exponent by one and differentiate constants to zero

\[\frac{d}{dx}\left(ax^{n}\right)=anx^{n-1}\]
  1. Rewrite roots and reciprocals Write roots as fractional powers and reciprocals as negative powers before applying the rule \(\displaystyle \sqrt{x}=x^{1/2},\quad \frac{1}{x^{2}}=x^{-2}\)
  2. Multiply the coefficient and exponent For a term ax^n, the new coefficient is a multiplied by n. \(\displaystyle 5x^{3}\quad\Rightarrow\quad 5\times3x^{2}=15x^{2}\)
  3. Reduce the exponent by one After multiplying the coefficient, replace n by n-1 in the power \(\displaystyle x^{n}\quad\Rightarrow\quad nx^{n-1}\)
  4. Drop constant terms A constant has no change as x changes, so its derivative is zero \(\displaystyle \frac{d}{dx}\left(-7\right)=0\)

Worked example

Differentiate the function shown and find its derivative at the given input

\[g(x)=4\sqrt{x}+\frac{6}{x^{2}}-3x+8,\quad x=1\]
  1. \(\displaystyle g(x)=4x^{1/2}+6x^{-2}-3x+8\)Rewrite the root and reciprocal as powers of x
  2. \(\displaystyle g'(x)=2x^{-1/2}-12x^{-3}-3\)Apply the power rule to each variable term and give the constant derivative as zero
  3. \(\displaystyle g'(1)=2(1)^{-1/2}-12(1)^{-3}-3\)Substitute the given input into the derivative, rather than the original function
  4. \(\displaystyle g'(1)=2-12-3=-13\)Simplify the three derivative terms to obtain -13

Common mistakes

Not reducing the exponent
✕\(\displaystyle \frac{d}{dx}\left(3x^{2}\right)=6x^{2}\)✓\(\displaystyle \frac{d}{dx}\left(3x^{2}\right)=6x\)
The power rule multiplies by the original exponent and reduces that exponent by one
Not carrying the coefficient
✕\(\displaystyle \frac{d}{dx}\left(4x^{3}\right)=3x^{2}\)✓\(\displaystyle \frac{d}{dx}\left(4x^{3}\right)=12x^{2}\)
The coefficient 4 is multiplied by the exponent 3, giving 12
Differentiating a constant as one
✕\(\displaystyle \frac{d}{dx}\left(8\right)=1\)✓\(\displaystyle \frac{d}{dx}\left(8\right)=0\)
A constant does not change when x changes, so its derivative is zero

Difficulty levels

introDifferentiate positive integer powers and simple polynomial terms
standardDifferentiate negative integer powers, roots or fractional powers
challengeDifferentiate rational expressions or evaluate a derivative at a specified point
examDifferentiate a mixed function and use a derivative condition or factorised quadratic

Questions about this topic

Why are roots and reciprocals rewritten as powers?+

At inputs where the real power function is differentiable, roots and reciprocals can be written as powers and differentiated with the power rule. Check the original domain and any endpoint separately

What happens to a constant term when it is differentiated?+

It becomes zero because changing x does not change a constant

Why do I differentiate before substituting the input?+

The derivative is a different function that gives the gradient. Differentiate first, then substitute the input

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 5: Differentiation

In this chapter

Evaluate limits and test continuityGradient of a tangent from first principlesDifferentiate power functions and evaluate a derivativeExpand and factor polynomial expressionsPascal rows and binomial coefficient structure5A Limits, first principles and the power rule: Simplify and combine algebraic fractionsEvaluate a polynomial or recover a coefficient parameterSubstitute values into a formula5A Limits, first principles and the power rule: Solve a linear equationTest or construct a smooth piecewise joinRead and construct a gradient function from a function graphAnalyse a cubic graph or determine its ruleEvaluate and determine function rulesEquation of a tangent or normal lineRecover a polynomial rule from derivative and tangent conditionsFind the gradient of a line through two pointsFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a pair of simultaneous linear equationsDetermine parameters in a polynomial identity5C Tangents, normals and derivative conditions: Solve a linear equationDifferentiate natural exponential functions and solve simple derivative conditionsExponential growth and decay rate models5D Exponential derivatives and rate models: Simplify and combine algebraic fractions5D Exponential derivatives and rate models: Round to decimal places and significant figuresDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsGeometric optimisation with a trigonometric modelSinusoidal models and their rates of changeSolve a practical right-angled-triangle problemPerimeter and area of standard shapes5E Trigonometric differentiation and rates: Round to decimal places and significant figuresApproximate a cubic root using Newton's method