Differentials and Chain Rule
Table of Contents
1. Total Differential
Given function \(f(x,y,z)\), we have
\[ \text{d}f = f_x\text{d}x + f_y\text{d}y + f_z\text{d}z \]
In this way, these differentials encode how changes in \(x\), \(y\), and \(z\) affect the function \(f\). Notice that it is similar to our formula for linear approximations, and so they can also be thought of as a placeholder for the small variations \(\Delta x\), \(\Delta y\), and \(\Delta z\).
2. Chain Rule
We can also divide by something like \(\text{d}t\) to get a rate of change if they depend on some external parameter \(t\), such as when \(x=x(t)\), \(y=y(t)\), and \(z=z(t)\), to get the chain rule:
\[ \frac{\text{d}f}{\text{d}t} = f_x\frac{\text{d}x}{\text{d}t} + f_y\frac{\text{d}y}{\text{d}t} + f_z\frac{\text{d}z}{\text{d}t} \]
2.1. Proof
From our formula for linear approximation, we have that:
\[ \Delta f \approx f_x \Delta x + f_y \Delta y + f_z \Delta z \]
Dividing everything by a change in time \(\Delta t\), we have:
\[ \frac{\Delta f}{\Delta t} \approx f_x\frac{\Delta x}{\Delta t} +f_y \frac{\Delta y}{\Delta t} + f_z \frac{\Delta z}{\Delta t} \]
Now, as we take the limit as \(\Delta t\) tends to zero, we get the following equality which is the chain rule:
\[ \lim_{\Delta t \to 0}{\frac{\Delta f}{\Delta t}} = \frac{\text{d}f}{\text{d}t} = f_x\frac{\text{d}x}{\text{d}t} + f_y\frac{\text{d}y}{\text{d}t} + f_z\frac{\text{d}z}{\text{d}t} \]
3. Applications of Chain Rule
3.1. Product Rule
We can use multivariable calculus and the chain rule to justify the product rule. Given a function \(f=uv\), where \(u=u(t)\) and \(v=v(t)\), we have that:
\[ \frac{\text{d}(uv)}{\text{d}t} = f_u \frac{\text{d}u}{\text{d}t} + f_v \frac{\text{d}v}{\text{d}t} = v \frac{\text{d}u}{\text{d}t} + u \frac{\text{d}v}{\text{d}t} \]
3.2. Quotient Rule
Given a function \(g=\frac{u}{v}\), where \(u=u(t)\) and \(v=v(t)\), we have:
\[ \frac{\text{d}(\frac{u}{v})}{\text{d}t} = \frac{1}{v} \frac{\text{d}u}{\text{d}t} - \frac{u}{v^2} \frac{\text{d}v}{\text{d}t} \]
3.3. Multiple Intermediate Variables
If we have a function \(f(u,v)\) where we have \(u(x,y)\) and \(v(x,y)\), we now have two intermediate variables \(x\) and \(y\). If we actually calculate the chain rule out, we have the following formulas:
\begin{aligned} \frac{\partial f}{\partial u} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial u} \\ \frac{\partial f}{\partial v} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial v} \end{aligned}