If $z = x + D(x + y)$ and I let $g(x,y) = x + y$
Would I be right in saying that :
$z = x + f(g(x))$
and
$\frac{dz}{dx} = 1 + \frac{df}{dg}\cdot\frac{dg}{dx}$?
If $z = x + D(x + y)$ and I let $g(x,y) = x + y$
Would I be right in saying that :
$z = x + f(g(x))$
and
$\frac{dz}{dx} = 1 + \frac{df}{dg}\cdot\frac{dg}{dx}$?
I assume $D(.)$ is a function not a constant.
Then yes: $ \frac{\partial }{\partial x}z(x,y) = 1 + \frac{\partial }{\partial g}D(g(x,y)) \frac{\partial }{\partial x}g(x,y)$
I think you meant $z=x+D(g(x,y))$. If $D$ is a function of one variable, then you can talk about the partial with respect to $x$; you would be correct in saying that $\frac{\partial z}{\partial x} = 1 + \frac{dD}{dg}\frac{\partial g}{\partial x}.$ That is, \frac{\partial z}{\partial x} = 1 + D'(x+y) because $\frac{\partial g}{\partial x} = \frac{\partial}{\partial x}(x+y) = 1$.
Notice that for functions of one variable, we use $d$ and $dx$, but for functions of several variables you are talking about the partial derivative, $\partial$ and $\partial x$.