# Partial derivatives

### Partial derivatives

#### Lessons

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 $y$.
• Introduction
Partial Derivatives Overview:
a)
Introduction to Partial Derivatives
• Derivatives in terms of 1 variable
• Treating all other variables as constants
• An example

b)
Definition of Partial Derivatives
• Recalling the definition of derivative
• Two formal equations
• Won't be Using them (Too Complicated)

c)
Application of Partial Derivatives
• Finding the tangent slope of a trace
• Seeing if the function is increasing or decreasing

• 1.
Finding the Partial Derivatives
Find the first order partial derivatives of the following function:

$f(x,y) = 2x \ln (xy^2) + \frac{x}{y} - \sqrt{x^3}$

• 2.
Find the first order partial derivatives of the following function:

$h(s,t) =$ $\sin(e^{s^2t^3}) + \tan \frac{2}{t}$

• 3.
Find the first order partial derivatives of the following function:

$g(r,s) = r \ln \sqrt{r^2 + s^2 + rs}$

• 4.
Find the slope of the traces to $z= \sqrt{4-x^2-y^2}$ at the point $(1, \sqrt{2} )$.

• 5.
Find the slope of the traces to $z =$ $\sin(xy)$ at the point $(0, \frac{\pi}{2})$.

• 6.
Is the Function Increasing or Decreasing?
Determine if $f(x,y)=y^2 \cos (\frac{x}{y})$ is increasing or decreasing at the point $(\frac{\pi}{2},1)$ if:
a)
We allow $x$ to vary and hold $y$ fixed.

b)
We allow $y$ to vary and hold $x$ fixed.

• 7.
Determine if $f(x,y)=x^2+y^2+ \ln (xy)$ is increasing or decreasing at the point $(1, 2)$ if:
a)
We allow $x$ to vary and hold $y$ fixed.

b)
We allow $y$ to vary and hold $x$ fixed.