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Given a function $f(t, y(t))$, how can I express its derivative with respect to $t$ and $d f/{d t}$?

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Suppose $x = x(s,t)$, and $y = y(s,t)$, the chain rule for two variables is

\begin{align} \frac{\partial}{\partial s} f\big(x(s,t),y(s,t)\big) &= \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}\\ \\ \frac{\partial}{\partial t} f\big(x(s,t),y(s,t)\big) &= \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t} \end{align}

What happens if $x(s,t) = t$ and $y(s,t) = y(t)$?

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@F'OlaYinka Really? Any multivariable calculus textbook will have the construction. A good introductory textbook is Marsden and Tromba's Vector Calculus. –  Pragabhava Oct 16 '12 at 4:39

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