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A LevelMaths: PureOCRRevision Notes7. Differentiation7.3 Further Differentiation7.3.2 Differentiating Other Functions (Trig, ln & e etc)

Differentiating Other Functions (Trig, ln & e etc) (OCR A Level Maths: Pure)

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Roger

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Differentiating Other Functions (Trig, ln & e etc)

How do I differentiate common functions?

  • These are the common results
    • fraction numerator d over denominator d x end fraction left parenthesis x to the power of n right parenthesis equals n x to the power of n minus 1 end exponent
    • fraction numerator d over denominator d x end fraction left parenthesis straight e to the power of x right parenthesis equals straight e to the power of x
    • fraction numerator d over denominator d x end fraction left parenthesis a to the power of x right parenthesis equals a to the power of x ln space a for a greater than 0
    • fraction numerator d over denominator d x end fraction left parenthesis ln space x right parenthesis equals 1 over x
    • fraction numerator d over denominator d x end fraction left parenthesis sin space x right parenthesis equals cos space x
    • fraction numerator d over denominator d x end fraction left parenthesis cos space x right parenthesis equals negative sin space x
    • fraction numerator d over denominator d x end fraction left parenthesis tan space x right parenthesis equals sec squared space x
    • fraction numerator d over denominator d x end fraction left parenthesis cot space x right parenthesis equals negative cos ec squared space x
    • fraction numerator d over denominator d x end fraction left parenthesis sec space x right parenthesis equals sec space x tan space x
    • fraction numerator d over denominator d x end fraction left parenthesis cos ec space x right parenthesis equals negative cos ec space x cot space x

How do I differentiate exponentials and logarithms?

  • The two basic differentiation formulae are:

Diff Other Funct Illustr 1, AS & A Level Maths revision notes

  • From those basic formulae are derived these two additional formulae:

Diff Other Funct Illustr 2_forms, AS & A Level Maths revision notes

Diff Other Funct Illustr 2_derivs, AS & A Level Maths revision notes

  • And for exponentials more generally:

Diff Other Funct Illustr 3, AS & A Level Maths revision notes

  • This last formula can be derived from Formula 3 by using the chain rule

Examiner Tip

  • The formulae for some of these derivatives are not given in the formulae booklet – you need to know them.
  • The formulae for ekx and ln x are the ones you absolutely do need to know .
  • The other formulae can be derived from those two as shown above, and remember – the derivative of ln kx is 1 over x, NOT k over x !

Worked example

Diff Other Funct Example, AS & A Level Maths revision notes

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    Roger

    Author: Roger

    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.