This quiz tests the work covered in the lecture on Computing Partial Derivatives
and corresponds to Section 14.2 of the textbook Calculus: Single and Multivariable
(Hughes-Hallett, Gleason, McCallum et al.).

There are more web quizzes at Wiley, select Section 2. This quiz was the same as the
Section 1 quiz at 14/12/05.

Suppose $f\left(x,y\right)=3{x}^{2}+2x{y}^{3}+4{y}^{2}\phantom{\rule{0.3em}{0ex}}.$
Which one of the following statements is correct? Exactly one option must be correct)

Choice (a) is incorrect

Try again, remember
to treat the $y$’s
as a constant when differentiating with respect to
$x$ and to
treat the $x$’s
as a constant when differentiating with respect to
$y\phantom{\rule{0.3em}{0ex}}.$

Choice (b) is incorrect

Try again, remember
to treat the $y$’s
as a constant when differentiating with respect to
$x$ and to
treat the $x$’s
as a constant when differentiating with respect to
$y\phantom{\rule{0.3em}{0ex}}.$

Choice (c) is correct!

Choice (d) is incorrect

Try again, remember
to treat the $y$’s
as a constant when differentiating with respect to
$x$ and to
treat the $x$’s
as a constant when differentiating with respect to
$y\phantom{\rule{0.3em}{0ex}}.$

Suppose $f\left(x,y\right)={x}^{3}{e}^{xy}\phantom{\rule{0.3em}{0ex}}.$
Which one of the statements is correct? Exactly one option must be correct)

Choice (a) is correct!

You
have correctly used the product rule when differentiating with respect to
$x$ and
$\frac{\partial f}{\partial y}={x}^{3}x{e}^{xy}={x}^{4}{e}^{xy}$

Choice (b) is incorrect

Try again, you must use the product rule to differentiate with respect to
$x$ and
regard ${x}^{3}$
as a constant when you differentiate with respect to
$y\phantom{\rule{0.3em}{0ex}}.$

Choice (c) is incorrect

Try again, you have not
differentiated ${e}^{xy}$ correctly with
respect to either variable.

Choice (d) is incorrect

Try again, you must use the product rule to differential with respect to
$x$ and
regard ${x}^{3}$
as a constant when you differentiate with respect to
$y\phantom{\rule{0.3em}{0ex}}.$

Consider $z={\left({x}^{2}y+3x{y}^{3}\right)}^{4}\phantom{\rule{0.3em}{0ex}}.$
Which one of the following statements is correct? Exactly one option must be correct)

Choice (a) is incorrect

Try again, you are not differentiating the internal function correctly.

Choice (b) is correct!

You have used the
chain rule correctly.

Choice (c) is incorrect

Try again, you have not used the chain rule correctly.

Choice (d) is incorrect

Try
again, you are not using the chain rule correctly.

Suppose $f\left(x,y\right)=\frac{sinxy}{{x}^{2}{y}^{3}}\phantom{\rule{0.3em}{0ex}}.$
Which of one the following statements is correct? Reduce your answer
to a fraction in lowest terms. Exactly one option must be correct)