Partial Derivatives

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Objectives:

1.: Understand how to compute a partial derivative.
2.: Know what the partial derivative represents geometrically.

Recap Video

The following video recaps the main ideas of the section.

Problems

If $f(x,y) = y^2+6xy$ , then:

$f_x = \answer {6y},\quad f_y = \answer {2y+6x}.$

If $f(x,y) = \sin (xy)+yz+\frac {x}{z}$ , then:

$f_x = \answer {y\cos (xy)+\frac {1}{z}},\quad f_y = \answer {x\cos (xy)+z},\quad f_z = \answer {y-\frac {x}{z^2}}.$

If $f(x,y) = xe^{xy}$ , then:

$f_x = \answer {xye^{xy}+e^{xy}},\quad f_y = \answer {x^2e^{xy}}.$

If $f(x,y) = \ln (xy)$ , then

$\frac {\partial ^2 f}{\partial y \partial x} = \answer {0}.$

If $f(x,y) = y^2x^3+x^2e^y$ , then:

$f_{xx} = \answer {6xy^2+2e^y},$
$f_{xy} = \answer {6x^2y+2xe^y},$
$f_{yx} = \answer {6x^2y+2xe^y},$
$f_{yy} = \answer {2x^3+x^2e^y}.$

If $f(x,y) = xe^{\sin (yz)}$ , then $f_{yzxyzx} = \answer {0}$ .

Use Clairaut’s theorem to reorder the partials.

The contour map of function $z = f(x,y)$ is shown below.

The partial $f_x(P)$ is and $f_y(P)$ is .

What are the $z$ -values doing as you move in the $x$ and $y$ directions?

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Recap Video

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