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Differentiating Other Functions (Cambridge (CIE) A Level Maths): Revision Note

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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

How do I differentiate the reciprocal trigonometric functions?

  • The formulae for the derivatives of the reciprocal trigonometric functions are:

begin mathsize 22px style fraction numerator d over denominator d x end fraction left parenthesis sec x right parenthesis equals sec x tan x end style

fraction numerator size 22px d over denominator size 22px d size 22px x end fraction size 22px left parenthesis size 22px cosec size 22px x size 22px right parenthesis size 22px equals size 22px minus size 22px cosec size 22px x size 22px cot size 22px x

fraction numerator size 22px d over denominator size 22px d size 22px x end fraction size 22px left parenthesis size 22px cot size 22px x size 22px right parenthesis size 22px equals size 22px minus size 22px cosec to the power of size 22px 2 size 22px x

  • You can derive the derivatives for sec, cosec, and cot using the chain rule and the derivatives of the basic trigonometric functions 

 

Diff Rec Inv Trig Illustr 1, AS & A Level Maths revision notes

Examiner Tips and Tricks

All of the above are in the formula book, make sure you know how to find them. They will be particularly useful when working with identities and when integrating trigonometric functions.

Worked Example

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

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