14.5 Chain rule

Remember

• If you remember from calc1 chain rule was

$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}$

• in this section we extend it to more than one variable

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Chain Rule of multiple variables

• there are multiple versions that deal with a dif case

Case 1

• for $z=f(x,y)$ and y and z are functions of t

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

Case 2

• For $z=f(x,y)$ and $x=g(s,t)$ $y=g(s,t)$

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

• a tree to visualize the goingons

General Version