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State the first-principles definition of the derivative.
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All Flashcards in Topic 5.12
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5.12.18 cards
State the first-principles definition of the derivative.
f'(x) = lim_{h→0} (f(x+h) − f(x))/h.
Geometrically, what is (f(x+h) − f(x))/h?
The gradient of the chord (secant) joining the points at x and x+h.
Why can't you just put h = 0 in the difference quotient?
You get 0/0, which is undefined. Simplify so the bottom h cancels first, then let h → 0.
What does the chord become as h → 0?
The tangent at the point — so its gradient becomes the derivative f'(x).
Differentiate x² from first principles.
((x+h)² − x²)/h = (2xh + h²)/h = 2x + h → 2x as h → 0.
Differentiate x³ from first principles.
((x+h)³ − x³)/h = 3x² + 3xh + h² → 3x² as h → 0.
What does the phrase 'from first principles' tell you to do?
Start from the limit definition of the derivative — do NOT just quote the power rule.
In ((x+h)² − x²)/h, why does the bottom h cancel cleanly?
After expanding, the top is 2xh + h² = h(2x + h); every term has a factor of h.
5.12.28 cards
What does lim_{x→a} f(x) = L mean informally?
As x gets close to a (from either side), f(x) gets close to L — the value f is heading towards.
What does it mean for a function to be continuous at a point?
No jump, hole or break there — you can draw through it without lifting your pen. Formally lim_{x→a} f(x) = f(a).
How do you find the second derivative f''(x)?
Differentiate f(x) to get f'(x), then differentiate f'(x) again.
What does the second derivative tell you?
The rate of change of the gradient — how the slope itself is changing.
Write the second derivative in Leibniz notation.
d²y/dx² (read 'd-two-y by d-x-squared'); the same as f''(x).
Does d²y/dx² mean (dy/dx)²?
No — the 2's are notation for differentiating twice; nothing is being squared.
Find f''(x) if f(x) = x³ − 4x².
f'(x) = 3x² − 8x, then f''(x) = 6x − 8.
What is d³y/dx³ for y = 2x³ + x²?
dy/dx = 6x² + 2x, d²y/dx² = 12x + 2, d³y/dx³ = 12.
Topic 5.12 study notes
Full notes & explanations for First principles (HL only)
Math AA exam skills
Paper structures, command terms & tips
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