5. The Chain Rule For Partial Derivatives

Chain Rule - One Variable

Chain Rule

If $h(x)=f(g(x))$ then

$h^{\prime }(x)=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Remark: If $y=g(x)$, then Chain rule can be rewritten to

$\frac{dh}{dx}=\frac{df}{dy}\frac{dy}{dx}$

Chain Rule - Simplest Version

Theorem

If $z=f(x,y)$, $x=g(t)$, and $y=h(t)$, then

$\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$

Chain Rule - Two Independent Variables

Theorem

If $z=f(x,y)$, $x=g(s,t)$, and $y=h(s,t)$, then

$\frac{\partial z}{\partial s}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial s}$

and

$\frac{\partial z}{\partial t}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}$

Remark: The rule can be generalized to the case where $z=f(x_1,x_2,\dots ,x_n)$ and each $x_i$ depends on $m$ variables $t_1,\dots ,t_m$.