10 Partial derivatives, part 2

Partial derivatives of functions \(\mathbb{R}^n \to \mathbb{R}\)

In the following definition, recall the standard basis vectors of \(\mathbb{R}^n\): \[ \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad \ldots, \quad \mathbf{e}_n = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}. \]
Suppose we write the variables in \(\mathbb{R}^n\) as \(x_1,x_2,\ldots,x_n\), so a function \(f:\mathbb{R}^n \to \mathbb{R}\) takes the form \(z = f(x_1,x_2,\ldots,x_n)\).

Definition.

Let \(f:\mathbb{R}^n \to \mathbb{R}\) be a differentiable function and \((a_1,a_2,\ldots,a_n)\) a fixed point in \(\mathbb{R}^n\).

The partial derivative of \(f\) with respect to \(x_j\) at \((a_1,a_2,\ldots,a_n)\) is defined as \[ \frac{\partial f}{\partial x_j}(a_1,a_2,\ldots,a_n) = f'(a_1,a_2,\ldots,a_n)\mathbf{e}_j. \]
If we replace \((a_1,a_2,\ldots,a_n)\) with a variable point \((x_1,x_2,\ldots,x_n)\), we get the partial derivative functions \[ \frac{\partial f}{\partial x_j}: \mathbb{R}^n \to \mathbb{R}. \]

How do you compute partial derivatives?

To compute the partial derivative of a function \(f:\mathbb{R}^n \to \mathbb{R}\) with respect to \(x_j\), we treat all other variables as constants and differentiate with respect to \(x_j\) using the usual rules of differentiation.

Exercise 1: Practice computing partial derivatives

Compute all partial derivatives of the following functions:

\(f(x,y,z) = xy + yz^2 + xz\)
\(g(x,y,z) = z\sin{(xy^2+2z)}\)
\(h(x,y,z,w) = \displaystyle\frac{x+y+z}{w}\)

Partial derivatives and (total) derivatives

Theorem.

Suppose that \(f: \mathbb{R}^n\to \mathbb{R}\) is a differentiable function. Then, as a \(1\times n\) matrix, we have \[ f'(x_1,x_2,\ldots,x_n) = \begin{bmatrix} \displaystyle\frac{\partial f}{\partial x_1}(x_1,x_2,\ldots,x_n) & \displaystyle\frac{\partial f}{\partial x_2}(x_1,x_2,\ldots,x_n) & \cdots & \displaystyle\frac{\partial f}{\partial x_n}(x_1,x_2,\ldots,x_n) \end{bmatrix}. \]

Theorem.

Suppose that \(f: \mathbb{R}^n\to \mathbb{R}\) is a differentiable function and \((a_1,a_2,\ldots,a_n)\) is a fixed point in \(\mathbb{R}^n\). Then, the tangent approximation to the function at this point is given by

\[ L(x_1,x_2,\ldots,x_n) = \sum_{j=1}^n \frac{\partial f}{\partial x_j}(a_1,a_2,\ldots,a_n)(x_j-a_j) + f(a_1,a_2,\ldots,a_n). \]

Exercise 2: Practice computing derivatives and tangent approximations

(a) Computing derivatives

Compute the (total) derivatives of the following functions:

\(f(x,y,z) = xy + yz^2 + xz\)
\(g(x,y,z) = z\sin{(xy^2+2z)}\)
\(h(x,y,z,w) = \displaystyle\frac{x+y+z}{w}\)

(b) Computing tangent approximations

Comput the tangent approximation of the function \(f\) in part (a) at the point \((1,1,1)\).

Higher-order partial derivatives

If \(f: \mathbb{R}^n \to \mathbb{R}\) is a differentiable function, then we have the partial derivative functions \[ \frac{\partial f}{\partial x_j}: \mathbb{R}^n \to \mathbb{R}, \quad j=1,2,\ldots,n. \]
These functions may themselves be differentiable, in which case we can take their partial derivatives.

Definition.

Let \(f:\mathbb{R}^n \to \mathbb{R}\) be a differentiable function. The second-order partial derivatives of \(f\) are the partial derivatives of the first-order partial derivatives, i.e. \[ \frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial}{\partial x_i}\left(\frac{\partial f}{\partial x_j}\right). \]

These may or may not exist.

A mixed second-order partial derivative is a second-order partial derivative of the form \(\displaystyle\frac{\partial^2 f}{\partial x_i \partial x_j}\) with \(i \neq j\).
A pure second-order partial derivative is a second-order partial derivative of the form \(\displaystyle\frac{\partial^2 f}{\partial x_j^2}\).

Theorem (Equality of mixed second-order partial derivatives).

Suppose that \(f:\mathbb{R}^n \to \mathbb{R}\) is a function whose second-order partial derivatives are continuous. Then, for all \(i,j = 1,2,\ldots,n\), we have

\[ \frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}. \]

Third and higher order partial derivatives are defined analogously, and the equality of mixed partial derivatives holds for all orders, provided that the relevant partial derivatives are continuous.

Exercise 3: Practice computing higher-order partial derivatives

Compute all third-order partial derivatives of the function

\[ f: \mathbb{R}^2 \to \mathbb{R}, \quad f(x,y) = 3xy^2-2y+5x^2y^2. \]