
We can find the gradient of PM and of PN. The gradient at P should lie between these two values. We can also find the gradient of MN which may give an even more accurate estimate.
This approach is called "differentiation from first principles"
This method may not be be part of your examination, but is here to give you a chance to see what lies behind the methods you use.
Summary/Background


Software/Applets used on this page
This question appears in the following syllabi:
Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|
AP Calculus AB (USA) | 3 | Differentiation | First principles | - |
AP Calculus BC (USA) | 3 | Differentiation | First principles | - |
AQA A-Level (UK - Pre-2017) | C1 | Differentiation | First principles | - |
AQA AS Maths 2017 | Pure Maths | Differentiation | From First Principles | - |
AQA AS/A2 Maths 2017 | Pure Maths | Differentiation | From First Principles | - |
CBSE XI (India) | Calculus | Limits and Derivatives | Definition of a derivative | - |
CCEA A-Level (NI) | C1 | Differentiation | First principles | - |
CIE A-Level (UK) | P1 | Differentiation | First principles | - |
Edexcel A-Level (UK - Pre-2017) | C1 | Differentiation | First principles | - |
Edexcel AS Maths 2017 | Pure Maths | Differentiation | From First Principles | - |
Edexcel AS/A2 Maths 2017 | Pure Maths | Differentiation | From First Principles | - |
I.B. Higher Level | 6 | Differentiation | First principles | - |
I.B. Standard Level | 6 | Differentiation | First principles | - |
Methods (UK) | M8 | Differentiation | First principles | - |
I.B. (MSSL) | 7 | Differentiation | First principles | - |
OCR A-Level (UK - Pre-2017) | C1 | Differentiation | First principles | - |
OCR AS Maths 2017 | Pure Maths | Differentiation Basics | From First Principles | - |
OCR MEI AS Maths 2017 | Pure Maths | Differentiation Basics | From First Principles | - |
OCR-MEI A-Level (UK - Pre-2017) | C2 | Differentiation | First principles | - |
Pre-U A-Level (UK) | 4 | Differentiation | First principles | - |
Scottish Advanced Highers | M1 | Differentiation | First principles | - |
Scottish (Highers + Advanced) | HM1 | Differentiation | First principles | - |
Scottish Highers | M1 | Differentiation | First principles | - |
Universal (all site questions) | D | Differentiation | First principles | - |
WJEC A-Level (Wales) | C1 | Differentiation | First principles | - |