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Partial Derivatives

Partial Derivatives: Formal Definition​

Given a function f:Rn→Rf: \mathbb{R}^n \rightarrow \mathbb{R}, the partial derivative of ff with respect to the ithi^{th} variable xix_i is defined as:

∂f∂xi=lim⁡Δxi→0f(x1,…,xi+Δxi,…,xn)−f(x1,…,xi,…,xn)Δxi\frac{\partial f}{\partial x_i} = \lim_{{\Delta x_i \to 0}} \frac{f(x_1, \ldots, x_i + \Delta x_i, \ldots, x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{\Delta x_i}

provided the limit exists. Here, Δxi\Delta x_i is a small change in the ithi^{th} variable while keeping all other variables constant.

Notation​

The partial derivative of ff with respect to xix_i is commonly denoted by ∂f∂xi\frac{\partial f}{\partial x_i} or fxif_{x_i}. When ff is a function of more than two variables, the notation becomes particularly useful to specify which variable we are differentiating with respect to.

Multivariate Functions​

For functions f:Rn→Rmf: \mathbb{R}^n \rightarrow \mathbb{R}^m where m>1m > 1, the partial derivatives can be organized into the Jacobian matrix:

J=(∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn)\mathbf{J} = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix}

Figures to add later...​

A potential figure illustrating partial derivatives could involve a 3D plot of a function f(x,y)f(x, y). The figure could show tangent planes corresponding to the partial derivatives ∂f∂x\frac{\partial f}{\partial x} and ∂f∂y\frac{\partial f}{\partial y} at a specific point. These planes would illustrate how the function changes when you vary only one variable while keeping the other constant.