Given \(z = f(x,y)\) such that \(x\) and \(y\) are themselves multi-variable functions of some parameters \(s\) and \(t,\)
\(\displaystyle
\frac{\mathrm{d}z}{\mathrm{d}s}
= \frac{\partial z}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}s}
+ \frac{\partial z}{\partial y} \frac{\mathrm{d}y}{\mathrm{d}s}
\)
\(\displaystyle
\frac{\mathrm{d}z}{\mathrm{d}t}
= \frac{\partial z}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}t}
+ \frac{\partial z}{\partial y} \frac{\mathrm{d}y}{\mathrm{d}t}
\,.
\)
Given a formula \(F(x,y)=0\) in which \(y\) is implicitly defined as a function of \(x,\)
\(\displaystyle
\frac{\mathrm{d}y}{\mathrm{d}x}
= -\dfrac{\dfrac{\partial F}{\partial x}}{\dfrac{\partial F}{\partial y}}
= -\frac{F_x}{F_y}
\,.
\)
Given a formula \(F(x,y,z)=0\) in which \(z\) is implicitly defined as a function of \(x\) and \(y\)
\(\displaystyle
\frac{\mathrm{d}z}{\mathrm{d}x}
= -\dfrac{\dfrac{\partial F}{\partial x}}{\dfrac{\partial F}{\partial z}}
= -\frac{F_x}{F_z}
\,.
\)
\(\displaystyle
\frac{\mathrm{d}z}{\mathrm{d}y}
= -\dfrac{\dfrac{\partial F}{\partial y}}{\dfrac{\partial F}{\partial z}}
= -\frac{F_y}{F_z}
\,.
\)
Collectively, these previous equations are known as the implicit function theorem.