MathNet AP: 3 - Differentiation - By first principles

Page ID: 1067

3 | Differentiation | e^x | «By first principles»

The 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

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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:

Syllabus	Module	Section	Topic	Exam Year
AP Calculus AB (USA)	3	Differentiation	e^x	-
AP Calculus BC (USA)	3	Differentiation	e^x	-
AQA A-Level (UK - Pre-2017)	C3	Differentiation	e^x	-
AQA A2 Maths 2017	Pure Maths	Differentiation	Differentiating Exponentials	-
AQA AS/A2 Maths 2017	Pure Maths	Differentiation	Differentiating Exponentials	-
CCEA A-Level (NI)	C3	Differentiation	e^x	-
CIE A-Level (UK)	P2	Differentiation	e^x	-
Edexcel A-Level (UK - Pre-2017)	C3	Differentiation	e^x	-
Edexcel A2 Maths 2017	Pure Maths	Differentiation	Differentiating Exponentials	-
Edexcel AS/A2 Maths 2017	Pure Maths	Differentiation	Differentiating Exponentials	-
I.B. Higher Level	6	Differentiation	e^x	-
I.B. Standard Level	6	Differentiation	e^x	-
Methods (UK)	M8	Differentiation	e^x	-
OCR A-Level (UK - Pre-2017)	C3	Differentiation	e^x	-
OCR A2 Maths 2017	Pure Maths	Differentiation Fundamentals	Differentiating Exponentials	-
OCR MEI A2 Maths 2017	Pure Maths	Differentiation Fundamentals	Differentiating Exponentials	-
OCR-MEI A-Level (UK - Pre-2017)	C3	Differentiation	e^x	-
Pre-U A-Level (UK)	4	Differentiation	e^x	-
Scottish Advanced Highers	M1	Differentiation	e^x	-
Scottish (Highers + Advanced)	AM1	Differentiation	e^x	-
Universal (all site questions)	D	Differentiation	e^x	-
WJEC A-Level (Wales)	C3	Differentiation	e^x	-