Let $g:\mathbb{R}\to\mathbb{R}$ and $h:\mathbb{R}^2\to\mathbb{R}$ be two differentiable functions.
I would like to compute the differential of $T:\mathbb{R}^2\to\mathbb{R}$ such that $T(x,y)=g(x+h(x,y))$.
I get $D T(x,y) = Dg(x+h(x,y))\circ D(x_1(\cdot)+h(\cdot)) = g'(x+h(x,y))\times \big(1+\partial_1 h(x,y) \quad \partial_2 h(x,y)\big)$.
Is it correct?