Free Implicit, related rates & first principles questions for IB Mathematics: Analysis & Approaches (SL & HL), HL only, part of Calculus. Each question has the answer and the marking points an examiner would look for.
Generate a full Implicit, Related Rates & First Principles worksheet →All Calculus questionsAir is pumped into a spherical balloon at a constant rate of 20 cm3 s−1.
Find the rate at which the radius is increasing when the radius is 5 cm.
| Part | Answer and marking points | Marks |
|---|---|---|
| 15π cm s−1 V = 43πr3 (A1) dVdr = 4πr2 A1 dVdt = dVdr × drdt M1 20 = 4π(5)2drdt A1 drdt = 15π A1 | 5 |
Use differentiation from first principles to show that the derivative of 1x is −1x2.
| Part | Answer and marking points | Marks |
|---|---|---|
| shown f′(x) = limh→0(1x + h − 1x) ÷ h M1A1 −hhx(x + h) A1 −1x(x + h) A1 as h → 0, f′(x) → −1x2 AG R1 | 5 |
A ladder of length 5 m leans against a vertical wall. The bottom of the ladder slides away from the wall at a constant 0.2 m s−1.
Find the rate at which the top of the ladder is sliding down the wall when the bottom is 3 m from the wall.
| Part | Answer and marking points | Marks |
|---|---|---|
| 0.150 m s−1 x2 + y2 = 25 (A1) 2xdxdt + 2ydydt = 0 M1A1 y = 4 when x = 3 (A1) dydt = −0.150, so sliding down at 0.150 m s−1 A1 | 5 |
A curve has equation x2y + 2y2 = 21.
Find an expression for dydx in terms of x and y.
Find the equation of the tangent to the curve at the point (−1, 3).
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | implicit derivative attempt at implicit differentiation M1 2xy + x2dydx + 4ydydx = 0 A1A1 rearranging for dydx A1 | 4 |
| (b) | y = 613x + 4513 substituting (−1, 3): dydx = 613 (M1)(A1) y = 613x + 4513 A1 | 3 |
Air is pumped into a spherical balloon at a constant rate of 50 cm3 s−1.
Find the rate at which the radius is increasing when the radius is 3 cm.
| Part | Answer and marking points | Marks |
|---|---|---|
| 2518π cm s−1 V = 43πr3 (A1) dVdr = 4πr2 A1 dVdt = dVdr × drdt M1 50 = 4π(3)2drdt A1 drdt = 2518π A1 | 5 |
Use differentiation from first principles to show that the derivative of x2 + 5x is 2x + 5.
| Part | Answer and marking points | Marks |
|---|---|---|
| shown f′(x) = limh→0(x + h)2 + 5(x + h) − x2 − 5xh M1A1 2xh + h2 + 5hh A1 2x + h + 5 A1 as h → 0, f′(x) → 2x + 5 AG R1 | 5 |
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Meet Riyaz →Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in