A LevelMathsCambridge (CIE)Pure 3Revision NotesDifferentiationFurther DifferentiationDifferentiating Other Functions

Differentiating Other Functions (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

Exam code: 9709

Written by: Amber

Reviewed by: Dan Finlay

Updated on 2 July 2026

Differentiating other functions

How do I differentiate common functions?

These are the common results
- $\frac{d}{d x} (x^{n}) = n x^{n - 1}$
- $\frac{d}{d x} (e^{x}) = e^{x}$
- $\frac{d}{d x} (a^{x}) = a^{x} \ln a$ for $a > 0$
- $\frac{d}{d x} (\ln x) = \frac{1}{x}$
- $\frac{d}{d x} (\sin x) = \cos x$
- $\frac{d}{d x} (\cos x) = - \sin x$
- $\frac{d}{d x} (\tan x) = \sec^{2} x$
- $\frac{d}{d x} (\cot x) = - \cos {ec}^{2} x$
- $\frac{d}{d x} (\sec x) = \sec x \tan x$
- $\frac{d}{d x} (\cos ec x) = - \cos ec x \cot x$

How do I differentiate e^kx, a^x, ln(kx) and a^kx?

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

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 a^kx:

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 reciprocal trigonometric functions?

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

$\frac{d}{d x} (\sec x) = \sec x \tan x$

$\frac{d}{d x} (cosec x) = - cosec x \cot x$

$\frac{d}{d x} (\cot x) = - {cosec}^{2} x$

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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Differentiating Other Functions (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

Differentiating other functions

How do I differentiate common functions?

How do I differentiate e^kx, a^x, ln(kx) and a^kx?

How do I differentiate reciprocal trigonometric functions?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Amber

Reviewer: Dan Finlay

Differentiating Other Functions (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

Differentiating other functions

How do I differentiate common functions?

How do I differentiate ekx, ax, ln(kx) and akx?

How do I differentiate reciprocal trigonometric functions?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Amber

Reviewer: Dan Finlay

How do I differentiate e^kx, a^x, ln(kx) and a^kx?