IBMathsDPAnalysis & Approaches (AA)SLRevision Notes5. CalculusFurther DifferentiationDifferentiating Special Functions

Differentiating Special Functions (DP IB Analysis & Approaches (AA)): Revision Note

Paul

Author

Paul

Last updated

DP Analysis & Approaches (AA): SLDP Analysis & Approaches (AA): HL

Did this video help you?

Differentiating Trig Functions

How do I differentiate sin and cos?

  • The derivative of space bold italic y equals bold sin space bold italic x is space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction equals bold cos space bold italic x   

  • The derivative of bold space bold italic y equals bold cos space bold italic x is space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction equals negative bold sin space bold italic x

  • For the linear functionbold space bold italic a bold italic x bold plus bold italic b, where bold italic a and bold italic b are constants,

    • the derivative of  bold space bold italic y bold equals bold sin bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis is space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold italic a bold cos bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis 

    • the derivative of bold space bold italic y bold equals bold cos bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis is bold space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold minus bold italic a bold sin bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis

  • For the general function space bold italic f bold left parenthesis bold italic x bold right parenthesis,

    • the derivative of bold space bold italic y equals bold sin stretchy left parenthesis bold italic f left parenthesis bold italic x right parenthesis stretchy right parenthesis is space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction equals bold italic f to the power of apostrophe left parenthesis bold italic x right parenthesis bold cos stretchy left parenthesis bold italic f open parentheses bold italic x close parentheses stretchy right parenthesis

    • the derivative of bold space bold italic y equals bold cos stretchy left parenthesis bold italic f left parenthesis bold italic x right parenthesis stretchy right parenthesis is space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction equals negative bold italic f to the power of bold italic apostrophe left parenthesis bold italic x right parenthesis bold sin stretchy left parenthesis bold italic f left parenthesis bold italic x right parenthesis stretchy right parenthesis

  • These last two sets of results can be derived using the chain rule

  • For calculus with trigonometric functions angles must be measured in radians

    • Ensure you know how to change the angle mode on your GDC

Examiner Tips and Tricks

  • As soon as you see a question involving differentiation and trigonometry put your GDC into radians mode

Worked Example

a) Find italic space f apostrophe left parenthesis x right parenthesis for the functions 

  1. space f left parenthesis x right parenthesis equals sin space x

  2. space f left parenthesis x right parenthesis equals cos space 2 x

  3. italic space f left parenthesis x right parenthesis equals 3 sin space 4 x minus cos left parenthesis 2 x minus 3 right parenthesis

5-2-1-ib-sl-aa-only-we1-soltn-a

b) Find the gradient of the tangent to the curve space y equals sin space stretchy left parenthesis 2 x plus straight pi over 6 stretchy right parenthesis space at the point where x equals straight pi over 8.

Give your answer as an exact value.

5-2-1-ib-sl-aa-only-we1-soltn-b

 

Did this video help you?

Differentiating e^x & lnx

How do I differentiate exponentials and logarithms?

  • The derivative of bold space bold italic y bold equals bold e to the power of bold italic x is bold space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold e to the power of bold italic x where x element of straight real numbers

  • The derivative of bold space bold italic y bold equals bold ln bold space bold italic x is bold space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold 1 over bold italic x where space x greater than 0

  • For the linear function bold space bold italic a bold italic x bold plus bold italic b, where a and b are constants,

    • the derivative of bold space bold italic y bold equals bold e to the power of bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis end exponent is text bold end text fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold italic a bold e to the power of bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis end exponent

    • the derivative of bold space bold italic y equals bold ln stretchy left parenthesis bold italic a bold italic x plus bold italic b stretchy right parenthesis is Error converting from MathML to accessible text.

      • in the special case space b equals 0, bold space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold 1 over bold italic x     (a's cancel)

  • For the general function bold space bold f bold left parenthesis bold italic x bold right parenthesis,

    • the derivative of  is Error converting from MathML to accessible text.

    • the derivative of  is 

  • The last two sets of results can be derived using the chain rule

Examiner Tips and Tricks

  • Remember to avoid the common mistakes:

    • the derivative ofspace ln space k x with respect to x isspace 1 over x, NOTspace k over x !

    • the derivative of straight e to the power of k x end exponent with respect to x is k straight e to the power of k x end exponent, NOT k x straight e to the power of k x minus 1 end exponent!

Worked Example

A curve has the equation size 16px space size 16px y size 16px equals size 16px e to the power of size 16px minus size 16px 3 size 16px x size 16px plus size 16px 1 end exponent size 16px plus size 16px 2 size 16px ln size 16px space size 16px 5 size 16px x.

Find the gradient of the curve at the point where space x equals 2 gving your answer in the form y equals a plus b e to the power of c, where a comma space b and c are integers to be found.

5-2-1-ib-sl-aa-only-we2-soltn
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It’s free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Test yourselfFlashcards

Did this page help you?

Previous:Increasing & Decreasing FunctionsNext:Chain Rule
Paul

Author: Paul

Expertise: Maths Content Creator

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

Download notes on Differentiating Special Functions