Partial derivatives

Notes:

Introduction to Partial Derivatives
Since we are dealing with multi-variable functions, we want to take the derivative in respect to 1 variable. This is known as the partial derivative.

For a function $f(x,y)$, we can have two partial derivatives:

• $f_x = \frac{df}{dx} \to$ derivative in terms of $x$
• $f_y = \frac{df}{dy} \to$ derivative in terms of $y$

When taking partial derivatives of functions in terms of a variable, we treat other variables as constants. In other words, we allow one variable to vary, and the other variables to be fixed.

Definition of Partial Derivatives
Recall from the definition of derivative, we have the formula:

$\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$

From for the definition of partial derivatives, we have the two following equations:

$f_x = \lim\limits_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}$
$f_y = \lim\limits_{h \to 0} \frac{f(x, y+h) - f(x,y)}{h}$

We won't be using these equations because we already know how to take derivatives.

Application of Partial Derivatives
We can use partial derivatives to find the tangent slope of the traces at a certain point $(a,b)$. We can do this by finding $f_x(a,b)$ and $f_y(a,b)$.
We can also use partial derivatives to see if $f(x,y)$ is increasing or decreasing. In other words, if

$f_x\gt0$, then $f(x,y)$ is increasing as we vary $x$.
$f_x\lt0$, then $f(x,y)$ is decreasing as we vary $x$.
$f_y\gt0$, then $f(x,y)$ is increasing as we vary $y$.
$f_y\lt0$, then $f(x,y)$ is decreasing as we vary $y$.