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first principlesThe displays below shows how the derivative of the exponential function e^x is found by using "differentiation from first principles". This version uses the \delta y, \delta x method, where \delta y means "a small increase in y" and \delta x means "a small increase in x".
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Summary/Background

The above display makes use of an important factor about the exponential function, namely that, as \delta x \to 0 so \displaystyle \frac{e^{\delta x} - 1}{\delta x} \to 1 .

Software/Applets used on this page

jsMath
This page uses jsMath
You can get a better display of the maths by downloading special TeX fonts from jsMath. In the meantime, we will do the best we can with the fonts you have, but it may not be pretty and some equations may not be rendered correctly.

Glossary

derivative

rate of change, dy/dx, f'(x), , Dx.

exponential function

A function having variables as exponents.

function

A rule that connects one value in one set with one and only one value in another set.

union

The union of two sets A and B is the set containing all the elements of A and B.

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopic
AP Calculus AB (USA)3Differentiationex
AP Calculus BC (USA)3Differentiationex
AQA A-Level (UK - Pre-2017)C3Differentiationex
AQA A2 Maths 2017Pure MathsDifferentiationDifferentiating Exponentials
AQA AS/A2 Maths 2017Pure MathsDifferentiationDifferentiating Exponentials
CCEA A-Level (NI)C3Differentiationex
CIE A-Level (UK)P2Differentiationex
G DifferentiationG2 Differentiating Further FunctionsDifferentiating e to x
Edexcel A-Level (UK - Pre-2017)C3Differentiationex
Edexcel A2 Maths 2017Pure MathsDifferentiationDifferentiating Exponentials
Edexcel AS/A2 Maths 2017Pure MathsDifferentiationDifferentiating Exponentials
I.B. Higher Level6Differentiationex
I.B. Standard Level6Differentiationex
Methods (UK)M8Differentiationex
OCR A-Level (UK - Pre-2017)C3Differentiationex
OCR A2 Maths 2017Pure MathsDifferentiation FundamentalsDifferentiating Exponentials
OCR MEI A2 Maths 2017Pure MathsDifferentiation FundamentalsDifferentiating Exponentials
OCR-MEI A-Level (UK - Pre-2017)C3Differentiationex
Pre-U A-Level (UK)4Differentiationex
Scottish Advanced HighersM1Differentiationex
Scottish (Highers + Advanced)AM1Differentiationex
Universal (all site questions)DDifferentiationex
WJEC A-Level (Wales)C3Differentiationex