TITLE: Chain rule for partial derivatives intuition
QUESTION [5 upvotes]: Can somebody give me an intuitive explanation for the below equations. I'm not sure how they come about and how they can be perceived logically.
$$\frac{\partial z}{\partial s} =\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial s} \ \ \text{and} \ \ \frac{\partial z}{\partial t} =\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial t} $$

REPLY [2 votes]: \begin{align}
z(s + \Delta s, t) &= f(x(s + \Delta s, t),y(s + \Delta s, t)) \\& \approx 
f \left(x(s,t) + \frac{\partial x(s,t)}{\partial s} \Delta s,y(s,t) + \frac{\partial y(s,t)}{\partial s} \Delta s \right) \\
\tag{$\spadesuit$}&\approx f(x(s,t),y(s,t)) + \frac{\partial f(x(s,t),y(s,t))}{\partial x} \frac{\partial x(s,t)}{\partial s} \Delta s
\\& \qquad \qquad \qquad \quad+ \frac{\partial f(x(s,t),y(s,t))}{\partial y}\frac{\partial y(s,t)}{\partial s} \Delta s.
\end{align}
Comparing this with
\begin{equation}
z(s + \Delta s, t) \approx z(s,t) + \frac{\partial z(s,t)}{\partial s} \Delta s
\end{equation}
we discover that
\begin{equation}
\frac{\partial z(s,t)}{\partial s} = 
\frac{\partial f(x(s,t),y(s,t))}{\partial x} \frac{\partial x(s,t)}{\partial s} 
+ \frac{\partial f(x(s,t),y(s,t))}{\partial y}\frac{\partial y(s,t)}{\partial s} .
\end{equation}
The key step is in line ($\spadesuit$), where we use the approximation
\begin{equation}
f(x + \Delta x, y + \Delta y)
\approx f(x,y) + \frac{\partial f(x,y)}{\partial x} \Delta x
+ \frac{\partial f(x,y)}{\partial y}{\Delta y}.
\end{equation}