Partial Derivatives

Since $f_x$ and $f_y$ are functions of $x$ and $y$ , we can take derivatives of these functions to get second derivatives. There are four such second derivatives, since each time we can differentiate with respect to $x$ or $y$ .

Notation for second partial derivatives

$\displaystyle \left(f_x\right)_x = f_{xx} = f_{11}= \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial ^2 f}{\partial x^2} = \frac{\partial ^2 z}{\partial x^2}$

$\displaystyle \left(f_x\right)_y = f_{xy} = f_{12} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial ^2 f}{\partial y \partial x} = \frac{\partial ^2 z}{\partial y \partial x}$

$\displaystyle \left(f_y\right)_x = f_{yx} = f_{21} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial ^2 f}{\partial x \partial y} = \frac{\partial ^2 z}{\partial x \partial y}$

$\displaystyle \left(f_y\right)_y = f_{yy} = f_{22} =\frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial ^2 f}{\partial y^2} = \frac{\partial ^2 z}{\partial y^2}$

The main result about higher derivatives is:

Clairaut's Theorem (or "mixed partials are equal"):
If

$f_{xy}$ and

$f_{yx}$ are both defined and continuous in a region containing the point

$(a,b)$ , then

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

A consequence of this theorem is that we don't need to keep track of the order in which we take derivatives. We just need to keep track of how many times we differentiate with respect to each variable.

Higher partial derivatives and Clairaut's theorem are explained in the following video.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Higher Order Derivatives