14.5 - Multivariable Chain Rule

Let’s start with an example equation for this concept.

Imagine an equation for modeling the perceived temperature, where several variables are responsible for predicting the temperature.

$z=f(T,H,W)$

Where:

• Z is the perceived temperature
• T is the temperature
• H is the humidity
• W is the wind chill

But what if the arguments themselves are determined by the other arguments? What if Humidity was determined by Temperature and Pressure?

$z=f(T,H(T,P),W(V,T))$

Where:

• P is pressure
• V is wind velocity

This is where the multivariable chain rule would come in handy.

---

Chain Rule: Case I

Lets make $z$ be defined by two parametric functions $x(t)$ and $y(t)$

$z=f(x,y)=f(x(t),y(t))$

Example

If $z=x^{2}y+3x^4$, and $x=\sin 2t$, $y=\cos t$, find $\frac{dz}{dt}$

Applying the chain rule:

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

Plugging in the numbers:

$\frac{dz}{dt}=(2xy+3y^4)(2{\cos }_{}2t)-(x^2+12x{y}^3)(\sin t)$

Simplifying:

$=z_x\frac{dx}{dt}{z}_y\frac{dy}{dt}$

---

Chain Rule: Algorithm

Let’s start with a tree diagram( should be familiar as a CS student)

Tree Diagram

graph TD
    z --> x
    z --> y
    x --> S["𝓈"]
    x --> t["t"]
    y --> s["𝓈"]
    y --> t2["t"]

How do we use this tree diagram?

Let’s take a look at an exmaple:

Example

If $W=f(x,y,,z)=x^{4}y+y^2{z}^3$, $x=rs{e}^t$, $y=r{s}^2{e}^{-}t$, $z=r^{2}s\sin \ast t$, find $\frac{dW}{dt}$

Applying the tree method:

\frac{dw}{ds} = w_{x}x_