Consider a differentiable function $f:\mathbb{R}^2 \mapsto \mathbb{R}$ with generic notation $f(x,y)$, and the function $h:\mathbb{R} \mapsto \mathbb{R}$ defined by $h(t)=f(x,t y)$ for some fixed $(x,y)$.
The function $h$ is itself differentiable. I have been fighting with colleagues over whether its derivative should be written $$h'(t)=y \dfrac{\partial f(x, t y)}{\partial y}$$ or $$h'(t)=y \dfrac{\partial f(x, t y)}{\partial (ty)}.$$
We have the same object in mind---the derivative of $f$ with respect to its second variable, evaluated at $(x,ty)$---but we disagree on which of these is the right notational convention. Which of these above is correct, or at least more standard?
Use the chain rule for $h(t)=f(x,ty)$ under the election of $g(t)=(x,ty)^{\top}$ giving $h'(t)=f'(g(t))\cdot g'(t)$. That is $$h'(t)=\frac{\partial f(x,ty)}{\partial y}y.$$
Addendo
I think it would be better to consider $G:\mathbb R^3\cdot\!\!\dashrightarrow\mathbb R^2$ via $$ \left(\begin{array}{c}t\\x\\y\end{array}\right) \longmapsto \left(\begin{array}{c}x\\ty\end{array}\right), $$ and $f:\mathbb R^2\cdot\!\!\dashrightarrow\mathbb R^1$ as $$\left(\begin{array}{c}v\\w\end{array}\right)\longmapsto f(v,w), $$ then $f(x,ty)=f\circ G(t,x,y)$.
So the chain rule would give that $$\left[\frac{\partial f(G)}{\partial t},\frac{\partial f(G)}{\partial x}, \frac{\partial f(G)}{\partial y}\right]$$ is equal the product of derivative $$\left[\frac{\partial f(G)}{\partial v},\frac{\partial f(G)}{\partial w}\right] \left(\begin{array}{ccc}0&1&0\\\\y&0&t\end{array}\right) = \left[y\frac{\partial f(G)}{\partial w},\frac{\partial f(G)}{\partial v},t \frac{\partial f(G)}{\partial w} \right] .$$ That is $$ \frac{\partial f(G)}{\partial t}=y\frac{\partial f(G)}{\partial w} \quad,\quad \frac{\partial f(G)}{\partial x}=\frac{\partial f(G)}{\partial v} \quad,\quad \frac{\partial f(G)}{\partial y}=t\frac{\partial f(G)}{\partial w}.$$