Differentiating Powers of x (AQA GCSE Further Maths)

Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Differentiating Powers of x

What is differentiation?

  • Differentiation is the process of finding an expression of the derivative (gradient function) from the equation of a curve

    • The equation of the curve is written y equals... and the gradient function is written fraction numerator straight d y over denominator straight d x end fraction equals...

How do I differentiate powers of x?

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

    • If y equals a x to the power of n then fraction numerator straight d y over denominator straight d x end fraction equals a n x to the power of n minus 1 end exponent

      • e.g.  If y equals 4 x cubed then fraction numerator straight d y over denominator straight d x end fraction equals 4 cross times 3 cross times x to the power of 3 minus 1 end exponent equals 12 x squared

      • you "bring down the power" then "subtract one from the power"

  • Don't forget these two special cases:

    • If y equals a x thenfraction numerator straight d y over denominator straight d x end fraction equals a

      • e.g.  If y equals 6 x then fraction numerator straight d y over denominator straight d x end fraction equals 6

    • If y equals a thenfraction numerator straight d y over denominator straight d x end fraction equals 0

      • e.g.  If y equals 5 then fraction numerator straight d y over denominator straight d x end fraction equals 0

    • These allow you to differentiate linear terms in x and constants

  • Functions involving fractions with denominators in terms of x will need to be rewritten as negative powers of x first

    • e.g.  If y equals 4 over x then rewrite as y equals 4 x to the power of negative 1 end exponent and differentiate

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

  •  The formulae for differentiating powers of x work for a sum or difference of powers of x

    • e.g.  Ifspace y equals 5 x to the power of 4 plus 3 x to the power of negative 2 end exponent plus 4 then
      fraction numerator d y over denominator d x end fraction equals 5 cross times 4 x to the power of 4 minus 1 end exponent plus 3 cross times open parentheses negative 2 close parentheses x to the power of negative 2 minus 1 end exponent plus 0
      fraction numerator d y over denominator d x end fraction equals 20 x cubed minus 6 x to the power of negative 3 end exponent

    • This is sometimes referred to differentiating 'term-by-term'

  • Products and quotients (divisions) cannot be differentiated in this way so they need expanding/simplifying first

    • e.g.  If y equals left parenthesis 2 x minus 3 right parenthesis left parenthesis x squared minus 4 right parenthesis then expand to y equals 2 x cubed minus 3 x squared minus 8 x plus 12 which is a sum/difference of powers of x and can then be differentiated

What can I do with derivatives (gradient functions)?

  • The derivative can be used to find the gradient of a function at any point

    • The gradient of a function at a point is equal to the gradient of the tangent to the curve at that point

    • A question may refer to the gradient of the tangent

Examiner Tips and Tricks

  • Don't try to do too many steps in your head; write the expression in a format that you can differentiate before you actually differentiate it

    • e.g. y equals 1 over x to the power of 4 plus 2 over x cubed can be rewritten as y equals x to the power of negative 4 end exponent plus 2 x to the power of negative 3 end exponent which is then far easier to differentiate

Worked Example

Find the derivative of 

(a)table row blank row blank row blank end table y equals 5 x cubed plus 2 x plus 3 over x squared plus 8

Rewrite the 3 over x squared term

y equals 5 x cubed plus 2 x plus 3 x to the power of negative 2 end exponent plus 8

Apply the rule for differentiating powers (y equals a x to the power of n comma space fraction numerator straight d y over denominator straight d x end fraction equals a n x to the power of n minus 1 end exponent) and apply the special cases for the terms 2 x and 8 (y equals a x comma fraction numerator space straight d y over denominator straight d x end fraction equals a and y equals a comma space fraction numerator straight d y over denominator straight d x end fraction equals 0)

fraction numerator straight d y over denominator straight d x end fraction equals 15 x squared plus 2 minus 6 x to the power of negative 3 end exponent

Unless a question specifies there is not usually a need to rewrite/simplify the answer

fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold 15 bold italic x to the power of bold 2 bold plus bold 2 bold minus bold 6 over bold italic x to the power of bold 3

 (b) y equals open parentheses 2 x plus 3 close parentheses squared

This is a product of two (equal) brackets so cannot be differentiated directly
Expand the brackets to get an expression in powers of x
Take time to get the expansion correct, writing stages out in full if necessary

table row y equals cell open parentheses 2 x plus 3 close parentheses open parentheses 2 x plus 3 close parentheses end cell row y equals cell 4 x squared plus 6 x plus 6 x plus 9 end cell row y equals cell 4 x squared plus 12 x plus 9 end cell end table

Differentiate 'term-by-term', looking out for those special cases

fraction numerator straight d y over denominator straight d x end fraction equals 8 x plus 12

There is a factor of 4 but there is no demand to factoise the final answer in the question

fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold 8 bold italic x bold plus bold 12

 

(c)table row blank row blank row blank end tabley equals fraction numerator 8 x to the power of 6 minus x cubed over denominator 2 x to the power of 4 end fraction

This is a quotient so cannot be differentiated directly
Spot the single denominator which means we can split the fraction by the two terms on the numerator

y equals fraction numerator 8 x to the power of 6 over denominator 2 x to the power of 4 end fraction minus fraction numerator x cubed over denominator 2 x to the power of 4 end fraction

Simplify using the laws of indices

y equals 4 x to the power of 6 minus 4 end exponent minus 1 half x to the power of 3 minus 4 end exponent
y equals 4 x squared minus 1 half x to the power of negative 1 end exponent

Each term is now a power of x, so differentiate 'term-by-term'

fraction numerator straight d y over denominator straight d x end fraction equals 8 x plus 1 half x to the power of negative 2 end exponent

There is demand to simplify or write the answer in a particular form

fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold 8 bold italic x bold plus bold 1 over bold 2 bold italic x to the power of bold minus bold 2 end exponent

Finding Gradients of Curves

Using the derivative to find the gradient of a curve

  • To find the gradient of a curve, at any point on the curve, substitute the x‑coordinate of the point into the derivative fraction numerator d y over denominator d x end fraction

Grad Tang Norm Illustr 1, A Level & AS Maths: Pure revision notes

Examiner Tips and Tricks

  • Read the question carefully; sometimes you are given fraction numerator d y over denominator d x end fraction and so don't need to differentiate initially - don't just automatically differentiate the first thing you see!

  • The following mean the same thing:

    • "Find the gradient of the curve at x equals 2"

    • "Find the gradient of the tangent at x equals 2"

      • the tangent gradient = curve gradient at that point

    • "Find the rate of change of y with respect to x at x equals 2"

Worked Example

A curve has the equation y equals 4 over 3 x cubed plus 3 x minus 8.

(a) Find the gradient of the curve when x equals 2.

y is already in a form that can be differentiated

fraction numerator straight d y over denominator straight d x end fraction equals 4 x squared plus 3

Substitute x equals 2 into fraction numerator straight d y over denominator straight d x end fraction

fraction numerator straight d y over denominator straight d x end fraction equals 4 cross times 2 squared plus 3 equals 19

The gradient of the curve at bold italic x bold equals bold 2 is 19

(b)

Work out the possible values of x for which the rate of change of y with respect to x is 4.

"Rate of change" is another way of describing the derivate

table row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals 4 row cell 4 x squared plus 3 end cell equals 4 end table

Solve this equation to find x
Note that it is quadratic equation so it could have up to two solutions
The question refers to 'values' implying there is (or could be) more than one value for x

table row cell 4 x squared end cell equals 1 row cell x squared end cell equals cell 1 fourth end cell row x equals cell plus-or-minus square root of 1 fourth end root end cell row x equals cell plus-or-minus 1 half end cell end table

The possible values of bold italic x, that give a rate of change of 4, are bold italic x bold equals bold 1 over bold 2 and bold italic x bold equals bold minus bold 1 over bold 2

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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.