# Higher order partial derivatives

### Higher order partial derivatives

#### Lessons

Notes:

2nd Order Partial Derivatives
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 2nd order partial derivatives here. For 2nd order partial derivatives, there are 4 types:

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

Where:
$f_{xx} \to$ 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 $x$

Higher Order Partial Derivatives
Of course, we can have even higher order partial derivatives. For example, we can have:

$f_{xxx} = \frac{d}{dx} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dx^3}$
$f_{xxy} = \frac{d}{dy} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dydx^2}$
$f_{xxxxx} = \frac{d}{dx} (\frac{d^4f}{dx^4}) = \frac{d^5f}{dx^5}$

We cannot list them all here because there is an infinite amount of higher order partial derivatives.

Clairaut's Theorem Suppose that $f$ is defined on a disk $D$ that contains the point $(a,b)$. If the functions $f_{xy}$ and $f_{yx}$ are continuous on this disk, then

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

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

b)
Higher Order Partial Derivatives
• Can go higher than 2nd order
• $f_{xxx}, f_{xxy}, f_{xxxxx}$
• An example

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

• 1.
Finding 2nd 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

$f(x,y) = x \tan \frac{x}{y} + e^{xy} + x^5$