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- Multivariable Calculus
- Partial Derivatives

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Get Started Now- Intro Lesson: a5:09
- Intro Lesson: b11:13
- Intro Lesson: c5:13
- Lesson: 16:19
- Lesson: 27:58
- Lesson: 39:28
- Lesson: 47:11
- Lesson: 55:42
- Lesson: 614:03

Since we can have higher order derivatives on a one variable function, we can also have this for multi-variable functions. We will specifically look at 2

$f_{xx} = \frac{d}{dx}(\frac{df}{dx}) = \frac{d^2f}{dx^2}$

$f_{xy} = \frac{d}{dy}(\frac{df}{dx}) = \frac{d^2f}{dydx}$

$f_{yy} = \frac{d}{dy}(\frac{df}{dy}) = \frac{d^2f}{dy^2}$

$f_{yx} = \frac{d}{dx}(\frac{df}{dy}) = \frac{d^2f}{dxdy}$

$f_{xx} \to$ derivative in respect to $x$ 2 times

$f_{yy} \to$ derivative in respect to $y$ 2 times

$f_{xy} \to$ derivative in respect to $x$ first, and then respect to $y$

$f_{yx} \to$ derivative in respect to $y$ first, and then respect to $x$

Of course, we can have even higher order partial derivatives. For example, we can have:

$f_{xxx} = \frac{d}{dx} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dx^3}$

$f_{xxy} = \frac{d}{dy} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dydx^2}$

$f_{xxxxx} = \frac{d}{dx} (\frac{d^4f}{dx^4}) = \frac{d^5f}{dx^5}$

$f_{xy}(a,b) = f_{yx}(a,b)$

- Introduction
**High Order Partial Derivatives Overview:**a)__2__^{nd}Order Partial Derivatives- 4 types of 2
^{nd}order partial derivatives - $f_{xx}, f_{xy}, f_{yy}, f_{yx}$
- An example

b)__Higher Order Partial Derivatives__

- Can go higher than 2
^{nd}order - $f_{xxx}, f_{xxy}, f_{xxxxx}$
- An example

c)__Clairaut's Theorem__- Two of the 2
^{nd}order partial derivatives are equal! - $f_{xy}(a,b) = f_{yx}(a,b)$
- An example to show they are equal

- 4 types of 2
- 1.
**Finding 2**^{nd}Order Partial Derivatives

Find all the second order derivatives for the following function$f(x,y) = x^3 y - \sqrt{4xy^3} + \ln (x^2)$

- 2.Find $f_{xx}$ and $f_{xy}$ for the following function
$f(x,y) = e^{x^2y^3} - \sin (x^2 + y^3)$

- 3.
**Finding Higher Order Partial Derivatives**

Given $w=e^{st}+ \sin (s^2)$, find $w_{ssstt}$ - 4.Given $f(x,y,z)={^4}\sqrt{(xyz)^3}$, find $\frac{d^4f}{dy^2dx^2}$
- 5.
**Verifying Clairaut's Theorem**

Verify Clairaut's Theorem for the given function$u(x,y) = \ln (x^2 - y)$

- 6.Verify Clairaut's Theorem for the given function
$f(x,y) = x \tan \frac{x}{y} + e^{xy} + x^5$