Step By Step Calculus » 11.6 - Related Rates

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Synopsis
Now we will consider the situation where xx and yy are related through such an equation g(x,y)=ag(x,y)=a but aa is no longer a constant but rather depend upon time. To find the rate of change \displaystyle \frac{da}{dt}\displaystyle \frac{da}{dt} of one item aa in terms of the rate of change \displaystyle \frac{dx}{dt}\displaystyle \frac{dx}{dt}, \dfrac{dy}{dt}\dfrac{dy}{dt} of other items xx, yy, we often derive an equation that relates these items, use the techniques of differentiation on the derived equation and then solve for the desired rate.
In the explicit case, we have g(x (t),y(t))=a(t)g(x (t),y(t))=a(t) as well as x(t)x(t), y(t)y(t), \dfrac{dx}{dt}\dfrac{dx}{dt}, and \displaystyle \frac{dy}{dt}\displaystyle \frac{dy}{dt} and we want \displaystyle \frac{da}{dt}\displaystyle \frac{da}{dt}. We find \displaystyle \frac{da}{dt}\displaystyle \frac{da}{dt} by substituting x(t)x(t), y(t)y(t), \dfrac{dx}{dt}\dfrac{dx}{dt}, and \displaystyle \frac{dy}{dt}\displaystyle \frac{dy}{dt} into the two dimensional chain rule:
\displaystyle \frac{da}{dt}=\frac{\partial}{\partial x}g(x(t),y(t))\frac{dx}{dt}+\frac{\partial}{\partial y}g(x(t),y(t))\frac{dy}{dt}.
\displaystyle \frac{da}{dt}=\frac{\partial}{\partial x}g(x(t),y(t))\frac{dx}{dt}+\frac{\partial}{\partial y}g(x(t),y(t))\frac{dy}{dt}.