%%{init: {'theme': 'base', 'themeVariables': { 'fontSize': '37px', 'fontFamily': 'arial' }}}%%
graph LR
X["X, Y, Z, T"]
B["Function f"]
Y["Y, W, V"]
X --> B
B --> Y
style B fill:#f3e5f5,stroke:#333,stroke-width:3px
style X fill:none,stroke:none
style Y fill:none,stroke:none
CS-466/566: Math for AI
Module 02: Multivariate Calculus
2026-03-23
Introduction to Derivatives
| Leibniz | Lagrange | Newton | Euler |
|---|---|---|---|
 \(\frac{dy}{dx}\) |
 \(y'(x)\) |
 \(\dot{y}\) |
 \(D_x y\) |
3. Product Rule Geometric Proof
Let \(A(x) = f(x)g(x)\) represent the area of a rectangle. A small change \(\Delta x\) increases this area by \(\Delta A(x)\), added in three regions:
- Right strip: \(g(x)[f(x+\Delta x) - f(x)]\)
- Top strip: \(f(x)[g(x+\Delta x) - g(x)]\)
- Corner: \([f(x+\Delta x) - f(x)][g(x+\Delta x) - g(x)]\) (vanishes as \(\Delta x \to 0\))
Total change in area: \[ \Delta A(x) = f(x)\Delta g + g(x)\Delta f + (\Delta f)(\Delta g) \]
Divide by \(\Delta x\) and take the limit: \[ A'(x) = f(x)g'(x) + g(x)f'(x) \]
Chain Rule Practice (1/2)
- Write the nested function in math notation: \(f(x) = \, ??\)
- Assume \(x=2\), compute \(u\), \(v\), and \(f\)
- Compute \(\frac{\partial f}{\partial x}\big|_{x=2}\) using the finite difference rule and the nested function notation you wrote. \[ \frac{df}{dx}\Big|_{x=a} = \frac{f(a+\triangle) + f(a-\triangle)}{2\triangle} \]
- Write the formula for the chain rule \(\frac{\partial f}{\partial x}\big|_{x=2}\) and use it to compute the derivative again and verify you get the same answer.
Multiple Inputs, Single Output
- Setup: \(n\) inputs (\(x_1, x_2, \ldots, x_n\)), one output \(f\).
- Question: How do we compute the derivative of the output with respect to each input?
Derivative of Functions with Multiple Inputs
- The derivative of the output with respect to the inputs is a vector of partial derivatives:
\[ \frac{\partial f}{\partial x} = \left[ \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n} \right] \]
- This vector is called the gradient of \(f\) with respect to \(x\). The output \(f\) changes by a small amount \(\Delta f\) when any input \(x_i\) changes by a small amount \(\Delta x_i\).
Example: Gradient of a Simple Function
Let \(f(x_1, x_2) = x_1^2 + x_2^2\).
The gradient is: \[
\nabla f = \left[ \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2} \right] = [2x_1, 2x_2]
\]
At point \((x_1, x_2) = (3, 4)\): \[\begin{align*} \frac{\partial f}{\partial x_1} &= 2x_1 = 2(3) = 6 \\ \frac{\partial f}{\partial x_2} &= 2x_2 = 2(4) = 8 \end{align*}\]
Numerical approximation at \((3,4)\):
- \(\frac{f(3.001, 4) - f(3, 4)}{0.001} \approx 6\)
- \(\frac{f(3, 4.001) - f(3, 4)}{0.001} \approx 8\)
Jacobian Matrix
Function with Multiple Inputs and Outputs:
For functions with multiple outputs, the derivative becomes a matrix called the Jacobian matrix (J):
\[\begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{bmatrix}\]
Thank You!
