Related rates of change

Watch a sliding ladder, a filling cone and a growing circle, pause at any moment, and see how one known rate fixes another through the chain rule.

Situation

Time t

Playback speed

Key equations

At this moment

Try this

Ladder: pause when the angle is 45° (x = y). Check that dy/dt = −dx/dt, then play on and see how fast the top moves as it nears the floor.
Cone: with a fixed inflow, compare dh/dt at h = 0.5 m and h = 1.00 m. Since r ∝ h, dh/dt ∝ 1/h², so it should be 4 times smaller.
Circle: double dr/dt and compare dA/dt at the same radius; notice that dC/dt stays constant while dA/dt keeps growing.

Common error: substituting the values at the moment (such as x = 3) before differentiating – differentiate the general relation with respect to t first, then substitute.

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More on 5: Chain and product rules · Second derivative and points of inflexion · Trapezoidal rule and Riemann sums · Displacement and distance travelled · Euler's method and differential equations · Implicit differentiation

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