DP IB Maths: AA HL

Revision Notes

5.2.1 Differentiating Special Functions

Test Yourself

Differentiating Trig Functions

How do I differentiate sin, cos and tan?

  • The derivative of is   
  • The derivative of is
  • The derivative ofbegin mathsize 16px style bold space bold italic y bold equals bold tan bold space bold italic x end style isbegin mathsize 16px style bold space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold sec to the power of bold 2 bold italic x end style
    • This result can be derived using quotient rule
  • 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 ofbegin mathsize 16px style 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 end style isbegin mathsize 16px style 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 end style 
    • the derivative ofbegin mathsize 16px style 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 end style isbegin mathsize 16px style 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 end style
    • the derivative ofbegin mathsize 16px style bold space bold italic y bold equals bold tan bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis end style isbegin mathsize 16px style bold space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold italic a bold sec to the power of bold 2 bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis end style
  • For the general functionbold space bold italic f bold left parenthesis bold italic x bold right parenthesis,
    • the derivative of is
    • the derivative of is
    • the derivative ofbegin mathsize 16px style bold space bold italic y bold equals bold tan bold left parenthesis bold italic f bold left parenthesis bold italic x bold right parenthesis bold right parenthesis end style isbegin mathsize 16px style bold space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold sec to the power of bold 2 bold left parenthesis bold italic f bold left parenthesis bold italic x bold right parenthesis bold right parenthesis end style
  • These last three 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

Exam Tip

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

Worked example

a)
Find 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 left parenthesis 5 x plus 1 right parenthesis

5-2-1-ib-hl-ai-aa-extraaa-we1a-soltn

b)       A curve has equationspace y equals tan space stretchy left parenthesis 6 x squared minus straight pi over 4 stretchy right parenthesis space.
Find the gradient of the tangent to the curve at the point where x equals fraction numerator square root of straight pi over denominator 2 end fraction.
Give your answer as an exact value.

5-2-1-ib-hl-ai-aa-extraaa-we1b-soltn-

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 0bold 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 
    • the derivative of  is 
  • The last two sets of results can be derived using the chain rule

Exam 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 equationspace y equals e to the power of negative 3 x plus 1 end exponent plus 2 ln space 5 x.

Find the gradient of the curve at the point wherespace x equals 2 giving 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 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.