The derivative as the gradient of a tangent and as a limit, f′(x) and dy/dx notation, the rule d/dx(xⁿ) = nxⁿ⁻¹ for any real n, derivatives of sin ax and cos ax, and sums and multiples.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
Calculus starts with one question: how steep is a curve at a single point? The answer — the derivative — is the most useful idea in all of mathematics.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Shrink h and watch the chord become the tangent.
Copy these into your notebook in your own words — that is how they stick.
f′(x) = dy/dx = lim_{h→0}[f(x + h) − f(x)]/h; gradient of the tangent.
d/dx(xⁿ) = nxⁿ⁻¹ for any real n.
d/dx(c) = 0; d/dx(cf) = c f′.
d/dx(f + g) = f′ + g′.
Rewrite: √x = x^{1/2}; 1/xⁿ = x⁻ⁿ before differentiating.
d/dx sin ax = a cos ax; d/dx cos ax = −a sin ax (radians).
Substitute x into dy/dx; set dy/dx = k to find where the gradient is k.
f′ > 0 increasing; f′ < 0 decreasing; f′ = 0 stationary.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (22 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [11 marks]
Answer all parts.
(a) Using first principles, show that the derivative of f(x) = x³ is 3x². [4]
(b) Differentiate y = 2x³ − 5x² + 3x − 1. [2]
(c) Differentiate y = 4√x + 3/x². [3]
(d) Differentiate y = 3 sin 2x + 2 cos 4x. [2]
Question 2 [11 marks]
f(x) = x³ − 3x.
(a) Find f′(x) and the gradient of the curve at x = 2. [3]
(b) Find the values of x at which the gradient of the curve is 9. [3]
(c) Find the coordinates of the points on the curve where the tangent is horizontal. [3]
(d) Write y = 2 − 1/x in a form suitable for differentiation and find dy/dx. [2]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.