# 8.2 Circular Motion: Position and Velocity Vectors

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## Chain Rule of Differentiation

Recall that when taking derivatives of a differentiable function $$f = f(\theta)$$ whose argument is also a differentiable function $$\theta=g(t)$$ then $$f=f(g(t))=h(t)$$ is a differentiable function of $$t$$ and

$$\frac{df}{dt}=\frac{df}{d\theta}\frac{d\theta}{dt}$$

Note that this is only in the case of circular motion, where $$r$$ is a constant in time. Otherwise, our derivative of $$f$$ would also have a $$\frac{df}{dr}\frac{dr}{dt}$$ term.

One way to measure an angle is in radians. A full circle has $$2\pi$$ radians. This week, we will use radians to measure the angles, so all angles will have units of radians, angular velocity will have units of radians/s, and angular acceleration will have units of radians/s$$^2$$. If we multiply these by a distance, such as $$r$$, the units will be m, m/s, or m/s$$^2$$.