Intuition for derivative of function in two variables, each of which is a function in a fourth variable.

Question

I'm trying to build an intuition for the partial derivative formula for: $$z = f(x, y), x = g(w), y = h(w)$$ where $z$ is a function in $x$ and $y$, and $x$ and $y$ themselves are each a function in $w$.

I know that: $$\frac{dz}{dw} = \frac{∂z}{∂x}\frac{dx}{dw} + \frac{∂z}{∂y}\frac{dy}{dw}$$ I also understand that the two terms in the sum are each an application of the chain rule. What I'm having trouble understanding is - why are the two terms being summed together?

Answer 1

The graph of a smooth function of two variables looks locally like a (slanted) plane lying above the $x$-$y$ plane.

Because of the slope, a slight increase in $x$ (say by $\varepsilon$) will result in a slight change in $z$ (by $\frac{\partial f}{\partial x}\cdot \varepsilon$ as a first order approximation). You can picture yourself walking on that surface, the derivative tells you how hard it is to walk up as you move in the direction of increasing $x$ values. Similarly an increase in $y$ will result in a change in $z$.

Now if you increase $x$ and then $y$, there are two successive changes in altitude which add up.

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If both $x$ and $y$ are considered as functions of some variable $w$, then an increase in $w$ (say by $\eta$) will result in a change of $x$ by $\frac{\partial x}{\partial w}\cdot \eta$ and this slight change in $x$ becomes your $\varepsilon$, hence the chain rule. The total change in $z$ is $$\frac{\partial f}{\partial x} \frac{\partial x}{\partial w}\cdot \eta +\frac{\partial f}{\partial y} \frac{\partial y}{\partial w}\cdot \eta $$

Answer 2

First of all, I assume the chain rule should have been written as $$ \frac{\mathrm{d}z}{\mathrm{d}w} = \frac{\partial f}{\partial x}\frac{\mathrm{d}g}{\mathrm{d}w} + \frac{\partial f}{\partial y}\frac{\mathrm{d}h}{\mathrm{d}w}. $$

Imagine the plane $x-y$ as a map where the function $f(x,y)$ denotes the altitude.

Then imagine a railway laid into the terrain described parametrically by a curve $x = g(w), y=h(w)$. One possible parametrisation is such that the parameter $w$ is a mileage.

The chain rule for $\frac{\mathrm{d}z}{\mathrm{d}w}$ then describes:

1. increment from moving in the $x$ direction: slope $\frac{\partial f}{\partial x}$ by movement $\frac{\mathrm{d}g}{\mathrm{d}w}$,
2. increment from moving in the $y$ direction: slope $\frac{\partial f}{\partial y}$ by movement $\frac{\mathrm{d}h}{\mathrm{d}w}$.

The other possible intuition is that:

1. Under sufficient condition the increment in altitude is $\nabla f \cdot \mathbf{s}$, where $\mathbf{s}$ is the velocity vector.
2. When you are travelling along the rails, the velocity (tangent vector) is $$ \mathbf{s} = [\frac{\mathrm{d}g}{\mathrm{d}w}, \frac{\mathrm{d}h}{\mathrm{d}w}] $$ which again yields the chain rule formula above.