Revision NotesPast Papers
IGCSEFurther MathsEdexcelRevision NotesCalculusApplications of DifferentiationFinding Gradients

Finding Gradients (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Finding Gradients

How do I find the gradient of a curve at a point?

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

    • Find the gradient by substituting the value of bold italic x at that point into the derivative of the curve

  • For example, if straight f open parentheses x close parentheses equals x squared plus 3 x minus 4

    • then straight f apostrophe open parentheses x close parentheses equals 2 x plus 3

    • the gradient of y equals straight f open parentheses x close parentheses when x equals 1 is  straight f to the power of apostrophe open parentheses 1 close parentheses equals 2 open parentheses 1 close parentheses plus 3 equals 5

    • the gradient of y equals straight f open parentheses x close parentheses when x equals negative 2 is  straight f to the power of apostrophe open parentheses negative 2 close parentheses equals 2 open parentheses negative 2 close parentheses plus 3 equals negative 1

  • Your exam calculator cannot find a derivative function in terms of x

    • But it may be able to find the numerical value of a derivative at a point

    • You can use this to check your work

Worked Example

A function straight f is defined by straight f open parentheses x close parentheses equals x cubed minus 6 x squared plus 5 x minus 12.

(a) Find straight f to the power of apostrophe open parentheses x close parentheses.

This is a 'powers of x' derivative

straight f to the power of apostrophe open parentheses x close parentheses equals 3 x to the power of 3 minus 1 end exponent minus 6 open parentheses 2 x to the power of 2 minus 1 end exponent close parentheses plus 5


bold f to the power of bold apostrophe stretchy left parenthesis x stretchy right parenthesis bold equals bold 3 bold italic x to the power of bold 2 bold minus bold 12 bold italic x bold plus bold 5

 

(b) Show that the curve y equals straight f open parentheses x close parentheses goes through the point open parentheses 2 comma space minus 18 close parentheses, and find the gradient of the tangent to the curve at that point.

Substitute x equals 2 into straight f open parentheses x close parentheses


table row cell straight f open parentheses 2 close parentheses end cell equals cell open parentheses 2 close parentheses cubed minus 6 open parentheses 2 close parentheses squared plus 5 open parentheses 2 close parentheses minus 12 end cell row blank equals cell 8 minus 24 plus 10 minus 12 end cell row blank equals cell negative 18 end cell end table



bold f stretchy left parenthesis 2 stretchy right parenthesis bold equals bold minus bold 18 so the curve goes through begin bold style stretchy left parenthesis 2 comma space minus 18 stretchy right parenthesis end style


Now substitute x equals 2 into straight f to the power of apostrophe open parentheses x close parentheses to find the gradient


table row cell straight f apostrophe open parentheses 2 close parentheses end cell equals cell 3 open parentheses 2 close parentheses squared minus 12 open parentheses 2 close parentheses plus 5 end cell row blank equals cell 12 minus 24 plus 5 end cell row blank equals cell negative 7 end cell end table


gradient bold equals bold minus bold 7

Last updated:

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Did this page help you?

    Previous:Techniques of DifferentiationNext:Stationary Points & Turning Points
    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    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.