Chain differentiation of bilinear form.

Question

Suppose one has a bilinear form $\alpha(x,y)$ and suppose one has the derivative $$\frac{\partial }{\partial t} \alpha(x,ty)$$ How does the chain differentiation work in this case? Is it $$\frac{\partial }{\partial (ty)} \alpha(x,ty) \frac{d}{dt}(ty)$$ or $$\frac{\partial }{\partial y} \alpha(x,ty) \frac{d}{dt}(ty)$$ As I understand it one should differentiate $\alpha$ with respect to the second argument so the first option is the correct one. But often I see the second option in books.