Introduction to partial derivatives

A function $z = f(x, y)$ depends on two independent variables, $x$ and $y$. A partial derivative measures how the function changes with respect to one variable, while holding the other constant.

Notation

1. $\frac{\partial z}{\partial x}$ or $f_x$ represents the partial derivative of $z$ with respect to $x$ (holding $y$ constant).
2. $\frac{\partial z}{\partial y}$ or $f_y$ represents the partial derivative of $z$ with respect to $y$ (holding $x$ constant).
3. Second partial derivatives:
   • $\frac{\partial^2 z}{\partial x^2}$ or $f_{xx}$ (derivative with respect to $x$ twice)
   • $\frac{\partial^2 z}{\partial y^2}$ or $f_{yy}$ (derivative with respect to $y$ twice)
   • $\frac{\partial^2 z}{\partial x \partial y}$ or $f_{xy}$ (derivative first with respect to $y$, then $x$)
   • $\frac{\partial^2 z}{\partial y \partial x}$ or $f_{yx}$ (derivative first with respect to $x$, then $y$)

Definition

The partial derivative of $z = f(x, y)$ with respect to $x$ is defined as:

$\frac{\partial z}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h}$

The partial derivative of $z = f(x, y)$ with respect to $y$ is defined as:

$\frac{\partial z}{\partial y} = \lim_{h \to 0} \frac{f(x, y + h) - f(x, y)}{h}$

Example 1: Find the first partial derivatives of $z = x^2 + xy + y^3$.

To find $\frac{\partial z}{\partial x}$, treat $y$ as a constant:

$\frac{\partial z}{\partial x} = 2x + y$

To find $\frac{\partial z}{\partial y}$, treat $x$ as a constant:

$\frac{\partial z}{\partial y} = x + 3y^2$

Example 2: Find the first partial derivatives of $z = \sin(3x) \cos(4y)$.

To find $\frac{\partial z}{\partial x}$, treat $y$ as a constant:

$\frac{\partial z}{\partial x} = 3\cos(3x) \cos(4y)$

To find $\frac{\partial z}{\partial y}$, treat $x$ as a constant:

$\frac{\partial z}{\partial y} = -4\sin(3x) \sin(4y)$

Example 3: Find $\frac{\partial^2 z}{\partial x^2}$ if $z = e^{3x + 2y}$.

First, find $\frac{\partial z}{\partial x}$:

$\frac{\partial z}{\partial x} = 3e^{3x + 2y}$

Now, differentiate with respect to $x$ again:

$\frac{\partial^2 z}{\partial x^2} = 9e^{3x + 2y}$