2nd order partial derivatives

Philippe Psaila Men working on the parabolic mirror array at Palma Del Rio, Spain

Concavity, and beyond!

Higher order derivatives

$$(f_x)_x=f_{xx}=\frac{\del}{\del x}\left( \frac{\del f}{\del x}\right)=\frac{\del^2 f}{\del x^2} =\frac{\del^2 z}{\del x^2}.$$ $$(f_{x})_{y}=f_{\color{blue}x\color{red}y} =\frac{\del}{\del y}\left( \frac{\del f}{\del x}\right) =\frac{\del^2 f}{\del {\color{red}y} \del {\color{blue}x}} =\frac{\del^2 z}{\del y \del x}.$$ The two different notations have $x$ and $y$ occurring in different orders. $$(f_y)_x=f_{yx}=\frac{\del}{\del x}\left( \frac{\del f}{\del y}\right)=\frac{\del^2 f}{\del x \del y} =\frac{\del^2 z}{\del x \del y}.$$ $$f_{yy}=(f_y)_y=\frac{\del}{\del y}\left( \frac{\del f}{\del y}\right)=\frac{\del^2 f}{\del y^2} =\frac{\del^2 z}{\del y^2}.$$

Interpretation

$f_x$ means...what? the slope in the $x$ direction

$f_{xx}$ means...what? It is the rate of change...of the $f_y$ as you movin in the $y$ direction--the concavity of the curve of a path in the $y$ direction

$f_{yx}$ means... the rate of change of the slope in the $y$ direction, as you increase $x$.

But is there a word like "concavity" to describe what that means graphically? See this visualization of $f_{yx}$ (Move myx$\equiv x$ to see how the $y$ slope changes with $x$.)

Example

Consider... $$\nonumber f (x, y) = x^3 + x^2y^3 – 2y^2$$

$$\nonumber f_x=3x^2+2xy^3;\ \ \ f_y=3y^2x^2-4y$$

$$\begineq f_{xx}=\frac{\del f_x}{\del x}=6x+2y^3; \ \ \ \ & f_{xy}=\frac{\del f_x}{\del y}=6xy^2\\ & f_{yx}=\frac{\del f_y}{\del x}=6xy^2;\ \ \ f_{yy}=\frac{\del f_y}{\del y}=6yx^2-4\\ \nonumber \endeq$$

Clairaut's theorem

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

Another way of saying this is $$\frac{\del}{\del y}\left(\frac{\del f}{\del x}\right) =\frac{\del}{\del x}\left(\frac{\del f}{\del y}\right).$$

See Clairaut's theorem and the meaning of $f_{xy}$.

To Do

• In the handouts folder... Go back to 10.2-3.partials.pdf. Do the last page: Mixed partials