Differentiation from first principles
Practice Flashcards
Flip to reveal answersState the first-principles definition of the derivative.
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All 8 Flashcards — Differentiation from first principles
Sign up free to track progress and get spaced-repetition review schedules.
Question
State the first-principles definition of the derivative.
Answer
f'(x) = lim_{h→0} (f(x+h) − f(x))/h.
Question
Geometrically, what is (f(x+h) − f(x))/h?
Answer
The gradient of the chord (secant) joining the points at x and x+h.
Question
Why can't you just put h = 0 in the difference quotient?
Answer
You get 0/0, which is undefined. Simplify so the bottom h cancels first, then let h → 0.
Question
What does the chord become as h → 0?
Answer
The tangent at the point — so its gradient becomes the derivative f'(x).
Question
Differentiate x² from first principles.
Answer
((x+h)² − x²)/h = (2xh + h²)/h = 2x + h → 2x as h → 0.
Question
Differentiate x³ from first principles.
Answer
((x+h)³ − x³)/h = 3x² + 3xh + h² → 3x² as h → 0.
Question
What does the phrase 'from first principles' tell you to do?
Answer
Start from the limit definition of the derivative — do NOT just quote the power rule.
Question
In ((x+h)² − x²)/h, why does the bottom h cancel cleanly?
Answer
After expanding, the top is 2xh + h² = h(2x + h); every term has a factor of h.
Read the notes
Full study notes for Differentiation from first principles
Topic 5.12 hub
First principles (HL only)
More from Topic 5.12
All flashcards in this topic
Math AA exam skills
Paper structures & tips
Track your progress with spaced repetition
Sign up free — Aimnova tells you exactly which cards to review and when, so you remember everything before your IB exam.
Start Free