Partial derivative of a function with another function inside it? What is
$\cfrac {\partial f(x, y, g(x))} {\partial x}$ expanded out?
I want to say $\cfrac {\partial f(x, y, g(x))} {\partial g(x)} \times \cfrac {\partial g(x)} {\partial x}$ but I don't think that's quite right.

$d_{x}f(x, y, g(x)) + \frac{dg}{dx} d_{z}f(x, y, g(x))$
f = f(x, y, z) and g(x) = f(x, y, g(x)). When x moves by dx, you are evaluating f to a new point where x AND z have changed.
(x, y, g(x)) -> (x+dx, y, g(x+dx)) ~ (x+dx, y, g(x) + g'(x)dx)
So when x moves a little bit, we will see a change in x and z.
Generally speaking, f(x+dx, y+dy, z+dz) ~ $f(x, y, z) + dx ∂_{x}f(x, y, z) + dy ∂_{y}f(x, y, z) + dz ∂_{z}f(x, y, z)$
So you replace dz by g'(x)dx and you will see the answer
