Partial derivatives turning into regular derivative

Question

If $z=z(x,y)$, and $x=x(t),y=y(t)$, write down the expression for $\frac{dz}{dx}$. I am confused by the difference between partial and regular derivatives when they are used together.

Answer 1

Observe that $$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$ $$\implies \frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dt}\cdot \frac{dt}{dx}$$ $$\implies \frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

Tell me if you don't get the first line.