The derivative of $f(t, y(t))$ with respect to $t$?

Question

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

Answer 1

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 $y(s,t) = y(t)$?