AP®MathsCollege BoardCalculus ABStudy GuidesUnit 2: Differentiation Definition & Fundamental PropertiesFundamental Properties of DifferentiationDerivative Rules

Derivative Rules (College Board AP® Calculus AB)

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Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Power rule

How do I differentiate powers of x?

  • Powers of x are differentiated according to the following formula:

    • Ifspace f left parenthesis x right parenthesis equals x to the power of n thenspace f apostrophe left parenthesis x right parenthesis equals n x to the power of n minus 1 end exponent 

    • This applies where r element of straight real numbers (real)

  • For example, if f open parentheses x close parentheses equals x to the power of 7

    • f apostrophe open parentheses x close parentheses equals 7 x to the power of 7 minus 1 end exponent equals 7 x to the power of 6

  • Be extra careful with fractional or negative powers, for example:

    • If g open parentheses x close parentheses equals x to the power of 2 over 3 end exponent then g apostrophe open parentheses x close parentheses equals 2 over 3 x to the power of 2 over 3 minus 1 end exponent equals 2 over 3 x to the power of negative 1 third end exponent

    • If h open parentheses x close parentheses equals x to the power of negative 3 end exponent then h apostrophe open parentheses x close parentheses equals negative 3 x to the power of negative 3 minus 1 end exponent equals negative 3 x to the power of negative 4 end exponent

  • This is much quicker than using the definition of a derivative, limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction

    • However you should still be able to use this definition to find a derivative

Worked Example

Find the derivative of the function f open parentheses x close parentheses equals x cubed,

(i) by using the power rule,

(ii) by using the definition of a derivative.

Answer:

(i) Ifspace f left parenthesis x right parenthesis equals x to the power of n thenspace f apostrophe left parenthesis x right parenthesis equals n x to the power of n minus 1 end exponent 

f apostrophe open parentheses x close parentheses equals 3 x to the power of 3 minus 1 end exponent equals 3 x squared

f apostrophe open parentheses x close parentheses equals 3 x squared

(ii) Use the definition of a derivative limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction

f open parentheses x close parentheses equals x cubed so this means f open parentheses x plus h close parentheses equals open parentheses x plus h close parentheses cubed

f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator open parentheses x plus h close parentheses cubed minus x cubed over denominator h end fraction

Expand the bracket and simplify

f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator x cubed plus 3 x squared h plus 3 x h squared plus h cubed minus x cubed over denominator h end fraction f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator 3 x squared h plus 3 x h squared plus h cubed over denominator h end fraction f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of open parentheses 3 x squared plus 3 x h plus h squared close parentheses

Consider what happens as h tends to zero

The terms containing an h will tend to zero

f apostrophe open parentheses x close parentheses equals 3 x squared

Derivatives of sums, differences and constant multiples

How do I differentiate sums and differences of powers of x?

  • When differentiating sums or differences of powers of x,

    • the derivative is simply the sum (or difference) of the derivatives of the terms

  • E.g. If f open parentheses x close parentheses equals x cubed plus x to the power of 7 minus x to the power of 12 plus x to the power of 1 half end exponent

    • Then f to the power of apostrophe open parentheses x close parentheses equals 3 x squared plus 7 x to the power of 6 minus 12 x to the power of 11 plus 1 half x to the power of negative 1 half end exponent

  • Note that products and quotients of powers of x cannot be differentiated in this way

    • They may need to be expanded or simplified first

How do I differentiate constant multiples of powers of x?

  • Constant multiples of powers of x are differentiated according to the following formula:

    • Ifspace f left parenthesis x right parenthesis equals a x to the power of n thenspace f to the power of apostrophe open parentheses x close parentheses equals a n x to the power of n minus 1 end exponent 

  • For example, if f open parentheses x close parentheses equals 12 x to the power of 4

    • f to the power of apostrophe open parentheses x close parentheses equals 12 cross times 4 x cubed equals 48 x cubed

  • Be careful with negative numbers

    • If g open parentheses x close parentheses equals negative 3 x to the power of negative 7 end exponent then g to the power of apostrophe open parentheses x close parentheses equals negative 3 cross times negative 7 x to the power of negative 8 end exponent equals 21 x to the power of negative 8 end exponent

  • This can then be applied to sums and differences

    • E.g. If h open parentheses x close parentheses equals 2 x to the power of 4 plus 7 x to the power of negative 5 end exponent minus 3 x to the power of 8

    • Then h to the power of apostrophe open parentheses x close parentheses equals 8 x cubed minus 35 x to the power of negative 6 end exponent minus 24 x to the power of 7

    • This is especially useful when differentiating polynomials

What special cases should I remember?

  • If f open parentheses x close parentheses equals a x then f to the power of apostrophe open parentheses x close parentheses equals a

    • E.g. If f open parentheses x close parentheses equals 4 x then f to the power of apostrophe open parentheses x close parentheses equals 4

    • This can also be seen graphically; the slope of the line y equals 4 x is 4

  • If g open parentheses x close parentheses equals a then g apostrophe open parentheses x close parentheses equals 0

    • E.g. If g open parentheses x close parentheses equals 2 then g to the power of apostrophe open parentheses x close parentheses equals 0

    • This can also be seen graphically; the line y equals 2 is horizontal so has a slope of 0

Worked Example

The function f open parentheses x close parentheses is defined by f open parentheses x close parentheses equals 2 x to the power of 3 over 2 end exponent plus 5 x to the power of negative 3 end exponent minus 9 x plus 7.

Find the derivative of f open parentheses x close parentheses.

Answer:

Differentiate each term individually and sum them together

Be especially careful with fractions and negatives

f to the power of apostrophe open parentheses x close parentheses equals 2 cross times 3 over 2 x to the power of 1 half end exponent space plus space 5 cross times negative 3 x to the power of negative 4 end exponent space minus space 9 space plus space 0

Simplify

f to the power of apostrophe open parentheses x close parentheses equals 3 x to the power of 1 half end exponent minus 15 x to the power of negative 4 end exponent minus 9

Simplifying expressions to find derivatives

How do I simplify an expression before differentiating?

  • If the function is not simply a sum of multiples of plus-or-minus x to the power of n, it may need to be simplified before it can be differentiated

  • You may need to expand

    • E.g. f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses open parentheses 2 x squared plus 5 close parentheses

    • Expand the brackets

      • f open parentheses x close parentheses equals 2 x cubed plus 4 x squared plus 5 x plus 10

    • This is now a sum of multiples of plus-or-minus x to the power of n

    • Differentiate each term

      • f to the power of apostrophe open parentheses x close parentheses equals 6 x squared plus 8 x plus 5

  • You may need to rewrite the expression as a power of x using laws of exponents

    • E.g. g open parentheses x close parentheses equals fraction numerator 9 x to the power of 4 cross times x cubed over denominator x squared end fraction

    • Simplify using laws of exponents

      • g open parentheses x close parentheses equals fraction numerator 9 x to the power of 7 over denominator x squared end fraction equals 9 x to the power of 5

      • This can now be differentiated

        • g to the power of apostrophe open parentheses x close parentheses equals 45 x to the power of 4

    • Also consider h open parentheses x close parentheses equals fraction numerator 2 over denominator square root of x end fraction

    • Rewrite using laws of exponents

      • h open parentheses x close parentheses equals 2 over x to the power of 1 half end exponent equals 2 x to the power of negative 1 half end exponent

    • This can now be differentiated

      • h to the power of apostrophe open parentheses x close parentheses equals negative x to the power of negative 3 over 2 end exponent

Worked Example

Differentiate the following functions.

(a) f open parentheses x close parentheses equals open parentheses 4 x minus 3 close parentheses open parentheses 6 x plus 2 over x close parentheses

Expand the brackets and simplify

f open parentheses x close parentheses equals 24 x squared plus 8 minus 18 x minus 6 over x

Rewrite the term 6 over x as a power of x

f open parentheses x close parentheses equals 24 x squared plus 8 minus 18 x minus 6 x to the power of negative 1 end exponent

Differentiate each term

f to the power of apostrophe open parentheses x close parentheses equals 48 x minus 18 plus 6 x to the power of negative 2 end exponent

(b) g open parentheses x close parentheses equals fraction numerator 2 x minus 6 square root of x over denominator x squared end fraction

Rewrite the square root as a power

g open parentheses x close parentheses equals fraction numerator 2 x minus 6 x to the power of 1 half end exponent over denominator x squared end fraction

Simplify using laws of exponents

g open parentheses x close parentheses equals fraction numerator 2 x over denominator x squared end fraction minus fraction numerator 6 x to the power of 1 half end exponent over denominator x squared end fraction equals 2 x to the power of negative 1 end exponent minus 6 x to the power of negative 3 over 2 end exponent

Differentiate each term

g to the power of apostrophe open parentheses x close parentheses equals negative 2 x to the power of negative 2 end exponent plus 9 x to the power of negative 5 over 2 end exponent

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    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.