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IBMaths: AA SLDPRevision Notes5. Calculus5.2 Further Differentiation5.2.1 Differentiating Special Functions

Differentiating Special Functions (DP IB Maths: AA SL)

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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 Tip

  • 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

 

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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 Tip

  • 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

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    Paul

    Author: Paul

    Expertise: Maths

    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 – one of the many reasons he is excited to be a member of the SME team.