Related rates

Related rates use the chain rule to connect time derivatives of quantities constrained by an equation. If $F(x(t),y(t))=0$, differentiating gives $F_x\dot{x}+F_y\dot{y}=0$. For a rectangle with one fixed side $a$ and variable side $b(t)$, its diagonal $d(t)=\sqrt{a^2+b(t{)}^2}$ obeys $\dot{d}=(b/d)\dot{b}$. The equation must be differentiated before inserting an instantaneous value of $b$, since that value does not make the changing side constant in time.