Differentiating Special Functions (Cambridge O Level Additional Maths)

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Differentiating Trig Functions

How do I differentiate trig functions?

  • For calculus with trigonometric functions, angles must be measured in radians
  • 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
  • The derivative of bold italic y bold equals bold tan bold space bold italic x is fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction equals bold sec to the power of bold 2 space bold italic x
  • The following two sets of relationships can be derived using the chain rule, but are useful to know!
  • 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
    • the derivative of 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 is 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
  • For the general function space bold 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 f left parenthesis bold italic x right parenthesis stretchy right parenthesis is Error converting from MathML to accessible text.
    • the derivative of bold space bold italic y equals bold cos stretchy left parenthesis bold 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 f to the power of bold apostrophe left parenthesis bold italic x right parenthesis bold sin stretchy left parenthesis bold f left parenthesis bold italic x right parenthesis stretchy right parenthesis
    • the derivative of bold space bold italic y bold equals bold tan stretchy left parenthesis 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 f to the power of bold apostrophe bold left parenthesis bold italic x bold right parenthesis sec squared stretchy left parenthesis f left parenthesis bold italic x right parenthesis stretchy right parenthesis

Examiner Tip

  • Remember that these rules only work in radians!

Worked example

a)
Find italic space straight f apostrophe left parenthesis x right parenthesis for the functions
i. space straight f left parenthesis x right parenthesis equals sin space x
ii.space straight f left parenthesis x right parenthesis equals cos space 2 x
ii.space straight f left parenthesis x right parenthesis equals 3 sin space 4 x minus cos left parenthesis 2 x minus 3 right parenthesis


i.
ii.    Use the chain rule or remember that when y equals cos open parentheses a x plus b close parentheses, then fraction numerator straight d y over denominator straight d x end fraction equals negative a sin open parentheses a x plus b close parentheses

iii.   Differentiate 'term by term'

straight f apostrophe open parentheses x close parentheses equals 3 open parentheses 4 cos 4 x close parentheses minus open parentheses negative 2 sin open parentheses 2 x minus 3 close parentheses close parentheses

Error converting from MathML to accessible text.

b)table row blank row blank end table
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.


Gradient of tangent is equal to gradient of curve. To find the gradient, differentiate...

fraction numerator straight d y over denominator straight d x end fraction equals 2 cos open parentheses 2 x plus pi over 6 close parentheses
... and substitute x equals pi over 8 into the derivative
fraction numerator straight d y over denominator straight d x end fraction equals 2 cos open parentheses 2 open parentheses pi over 8 close parentheses plus pi over 6 close parentheses
Ensure that your calculator is in radians

equals fraction numerator square root of 6 minus square root of 2 over denominator 2 end fraction equals 0.517638090...
Answer = fraction numerator square root of bold 6 bold minus square root of bold 2 over denominator bold 2 end fraction, or 0.518 to 3 significant figures
 

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
  • In addition, the results below can all be found using the chain rule but are worthwhile knowing, to save time in an exam
  • 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 space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction equals fraction numerator bold italic a over denominator stretchy left parenthesis bold italic a bold italic x plus bold italic b stretchy right parenthesis end fraction
      • 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 Error converting from MathML to accessible text. is 

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 space y equals straight e to the power of negative 3 x end exponent plus 2 ln space x.

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

Differentiate each term separately. 

fraction numerator straight d y over denominator straight d x end fraction equals negative 3 straight e to the power of negative 3 x end exponent space plus space 2 open parentheses 1 over x close parentheses

Substitute x space equals space 2 into the derivative.

fraction numerator straight d y over denominator straight d x end fraction equals negative 3 straight e to the power of negative 3 open parentheses 2 close parentheses end exponent space plus space 2 open parentheses 1 half close parentheses 

Rearrange to the required form.

table row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals cell negative 3 straight e to the power of negative 6 end exponent space plus space 1 space equals space 1 space minus space 3 straight e to the power of negative 6 end exponent end cell end table

Do not use your calculator to evaluate this as the question asks for the answer given in this form.

bold 1 bold space bold minus bold space bold 3 bold e to the power of bold minus bold 6 bold space end exponent

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.