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".

## 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

## 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.

## 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} | - |