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- Partial Derivatives

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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 $y$

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 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.