Defining and Differentiating Vector-Valued Functions BC only

A vector-valued function $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ is two parametric equations in one bracket, and $\mathbf{r}'(t) = \langle x'(t), y'(t)\rangle$ is taken one component at a time with every differentiation rule unchanged. The errors all come from letting the components interact: differentiating only one, combining them into $\sqrt{x^{2}+y^{2}}$ before differentiating, or swapping the slots on reassembly. Position, velocity and speed are three different types: the first two are vectors and the third is the scalar $|\mathbf{v}| = \sqrt{(dx/dt)^{2} + (dy/dt)^{2}}$, which is 9.3's arc length integrand exactly, and that identity is why distance travelled is the integral of speed.

Acceleration is $\mathbf{r}''$, componentwise, and it points where the velocity is changing rather than where the particle is going. Uniform circular motion makes the distinction concrete: $\mathbf{r} = \langle\cos t, \sin t\rangle$ has constant speed $1$ and acceleration $-\mathbf{r}$ pointing at the origin, so constant speed and nonzero acceleration coexist. Note also that $|\mathbf{a}|$ is not the derivative of $|\mathbf{v}|$, since taking a magnitude and differentiating do not commute. A velocity of $\mathbf{0}$ needs both components to vanish at once, and the velocity vector being tangent to the path recovers 9.1's slope as $y'(t)/x'(t)$.