Implicit differentiation

The idea

In a mechanism, the geometry ties positions together, e.g. “these two sides and this angle make a triangle”. You often can’t rearrange that into $y=f(x)$, and you don’t need to. Differentiate both sides as they stand, using the chain rule on anything that depends on time. That turns a position equation into a velocity equation in one step.

The rule

Anything that changes with time gets the chain rule: $\frac{d}{dt}\left(x^2\right)=2x\dot{x}\frac{d}{dt}\left(\cos \theta \right)=-\sin \theta \dot{\theta }\frac{d}{dt}\left(s^2\right)=2s\dot{s}$ Constants (fixed link lengths) differentiate to zero. Products need the product rule, which you’ll need again on the second differentiation: $\frac{d}{dt}\left(s\dot{s}\right)={\dot{s}}^2+s\ddot{s}$

The recipe for mechanisms

1. Write the geometry: Pythagoras, trig, or the cosine rule $c^2=a^2+b^2-2ab\cos C$.
2. Differentiate with respect to time → velocities ($\dot{s}$, $\dot{\theta }=\omega$).
3. Differentiate again → accelerations ($\ddot{s}$, $\ddot{\theta }=\alpha$).
4. Put in the numbers at the instant you care about. Watch the signs: a length that’s getting shorter has negative $\dot{s}$.

Worked example

A 5 m ladder leans against a wall. Its foot slides away from the wall at 0.5 m/s. How fast is the top sliding down when the foot is 3 m out?

Answer

Geometry: $x^2+y^2=25$. When $x=3$, $y=4$. Differentiate: $2x\dot{x}+2y\dot{y}=0$ → $\dot{y}=-\frac{x\dot{x}}{y}=-\frac{3\times 0.5}{4}=-0.375$ m/s The top slides down at 0.375 m/s.

This is exactly the same maths as a robot’s linear actuator pushing a link, or the A2 slider-crank.

Common mistakes

• Forgetting the $\dot{x}$ (the chain rule part) on each term.
• Substituting numbers before differentiating. Differentiate the general equation first.
• Using $\dot{s}=0$ when the rate is constant. It’s $\ddot{s}$ that’s zero.

Check yourself

1. Differentiate $y=0.2\cos \theta$ with respect to time, twice.

$\dot{y}=-0.2\sin \theta \dot{\theta }$, $\ddot{y}=-0.2\cos \theta {\dot{\theta }}^2-0.2\sin \theta \ddot{\theta }$.