)=(x(t),y(t))$), then
    how do we discover how do changes in $t$ affect $f$? In other
    words, what is the derivative of $f$ with respect to $t$?
    Notationally, we seek $\ds \frac{df}{dt}$ which we formally
    write as $\ds \frac{d}{dt}[f(x(t),y(t))]$ or $\ds \frac{d}{dt}
    [f(\vec r(t))].$
  </div>To answer this problem, we use the chain rule, which is
  just matrix multiplication.

  <div class="theorem" id="chain-rule-def"><b>The Chain Rule</b>
     Let $\vec x$ be a vector and $\vec f$
    and