CLASS 12 MATHS · APPLICATIONS OF DERIVATIVES

Rate of change using derivatives

If A depends on r and r changes with time, dA/dt = (dA/dr) × (dr/dt).

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Worked examples

Example 1 · level 1

Priya drops a stone into a still pond; the circular ripple's radius grows at 1 cm/s. How fast is the enclosed area increasing (in cm²/s) when the radius is 8 cm? The answer is a multiple of π: enter the number that multiplies π.

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dA/dt = 2πr × dr/dt = 2π × 8 × 1 = 16π cm²/s.
The coefficient of π is 16.

Answer: 16

Example 2 · level 2

A spherical balloon is being inflated for a school annual day party, and its radius increases at 1 cm/s. Find the rate at which its volume (cm³/s) increases when the radius is 5 cm. Enter the number that multiplies π.

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dV/dt = 4π × 25 × 1 = 100π cm³/s.
Coefficient of π = 100.

Answer: 100

Example 3 · level 3

Each edge of a metal cube expands at 2 cm/s when heated. How fast (in cm³/s) is the volume increasing when the edge is 7 cm?

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dV/dt = 3x² × dx/dt = 3 × 49 × 2 = 294 cm³/s.

Answer: 294