## H1011 Quizzes

Quiz 8: Partial derivatives
Question 1 Questions
Find the first order partial derivative with respect to $x$ of $f\left(x,y\right)={e}^{3x}cosy$.
 a) ${f}_{x}\left(x,y\right)=3{e}^{3x}cosy+{e}^{3x}siny$. b) ${f}_{x}\left(x,y\right)=3{e}^{3x}cosy+{e}^{3x}cosy$. c) ${f}_{x}\left(x,y\right)=-{e}^{3x}siny$. d) ${f}_{x}\left(x,y\right)=3{e}^{3x}cosy$.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Note that since we are differentiating with respect to $x$, we treat $cosy$ as a constant.
Find the first order partial derivative with respect to $y$ of $f\left(x,y\right)={x}^{3}{y}^{2}+3x{e}^{y}$.
 a) ${f}_{y}\left(x,y\right)=3{x}^{2}{y}^{2}+3{e}^{y}$ b) ${f}_{y}\left(x,y\right)=3{x}^{2}{y}^{2}+2{x}^{3}y+3{e}^{y}+3x{e}^{y}$ c) ${f}_{y}\left(x,y\right)=2{x}^{3}y+3x{e}^{y}$ d) ${f}_{y}\left(x,y\right)=6{x}^{2}y+3{e}^{y}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
We treat ${x}^{3}$ and $3x$ as constants and differentiate term by term with respect to $y$.
Choice (d) is incorrect
Find the second order partial derivatives of $f\left(x,y\right)={\left(3x+2y\right)}^{4}$.
 a) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 12{\left(3x+2y\right)}^{3}& \text{}\\ {f}_{yy}\left(x,y\right)& =& 24{\left(3x+2y\right)}^{2}& \text{}\end{array}$ b) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 36{\left(3x+2y\right)}^{2}& \text{}\\ {f}_{yy}\left(x,y\right)& =& 8{\left(3x+2y\right)}^{3}& \text{}\end{array}$ c) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 24{\left(3x+2y\right)}^{2}& \text{}\\ {f}_{xy}\left(x,y\right)& =& 32\left(3x+2y\right)& \text{}\end{array}$ d) $\begin{array}{rcll}{f}_{xy}\left(x,y\right)& =& 108{\left(3x+2y\right)}^{2}& \text{}\\ {f}_{yy}\left(x,y\right)& =& 48{\left(3x+2y\right)}^{2}& \text{}\end{array}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
${f}_{x}\left(x,y\right)=12{\left(3x+2y\right)}^{3},{f}_{y}\left(x,y\right)=8{\left(3x+2y\right)}^{3},$
${f}_{xx}\left(x,y\right)=108{\left(3x+2y\right)}^{2},{f}_{yy}\left(x,y\right)=48{\left(3x+2y\right)}^{2}.$
Find the second order partial derivatives of $f\left(x,y\right)=3{x}^{2}y+4cosxy$.
 a) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 6y-4{y}^{2}cosxy& \text{}\\ {f}_{yy}\left(x,y\right)& =& -4{x}^{2}cosxy& \text{}\end{array}$ b) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& -4{x}^{2}cosxy& \text{}\\ {f}_{yy}\left(x,y\right)& =& 6y-4{y}^{2}cosxy& \text{}\end{array}$ c) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 6x-4{x}^{2}cosxy& \text{}\\ {f}_{yy}\left(x,y\right)& =& -4{y}^{2}cosxy& \text{}\end{array}$ d) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 6x-4xycosxy& \text{}\\ {f}_{yy}\left(x,y\right)& =& -4xycosxy& \text{}\end{array}$

Choice (a) is correct!
${f}_{x}\left(x,y\right)=6xy-4ysinxy,{f}_{y}\left(x,y\right)=3{x}^{2}-4xsinxy$,
${f}_{xx}\left(x,y\right)=6y-4{y}^{2}cosxy,{f}_{yy}\left(x,y\right)=-4{x}^{2}cosxy.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
What is the mixed, second order partial derivative of $f\left(x,y\right)=\sqrt{2{x}^{2}+{y}^{2}}$ ?
 a) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 2{\left(2{x}^{2}+{y}^{2}\right)}^{-1∕2}-4{x}^{2}{\left(2{x}^{2}+{y}^{2}\right)}^{-3∕2}& \text{}\\ {f}_{yy}\left(x,y\right)& =& {\left(2{x}^{2}+{y}^{2}\right)}^{-1∕2}-{y}^{2}{\left(2{x}^{2}+{y}^{2}\right)}^{-3∕2}& \text{}\end{array}$ b) ${f}_{xy}\left(x,y\right)=-2xy{\left(2{x}^{2}+{y}^{2}\right)}^{-3∕2}$ c) ${f}_{xy}\left(x,y\right)=-4xy{\left(2{x}^{2}+{y}^{2}\right)}^{-1∕2}$ d) ${f}_{xy}\left(x,y\right)=-4{x}^{2}{\left(2{x}^{2}+{y}^{2}\right)}^{-3∕2}$

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{rcll}{f}_{x}\left(x,y\right)& =& 2x{\left(2{x}^{2}+{y}^{2}\right)}^{-1∕2}& \text{}\\ {f}_{xy}\left(x,y\right)& =& -2xy{\left(2{x}^{2}+{y}^{2}\right)}^{-3∕2}& \text{}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
If $f\left(x,y\right)={x}^{3}-2xy+x{y}^{3}+3{y}^{2}$ which of the following is true ?
 a) $\begin{array}{rcll}{f}_{x}\left(x,y\right)& =& 3{x}^{2}-2y+{y}^{3}& \text{}\\ {f}_{y}\left(x,y\right)& =& -2x+6y+3x{y}^{2}& \text{}\\ {f}_{xy}\left(x,y\right)& =& 6x& \text{}\end{array}$ b) $\begin{array}{rcll}{f}_{x}\left(x,y\right)& =& -2x+6y+3x{y}^{2}& \text{}\\ {f}_{y}\left(x,y\right)& =& 3{x}^{2}-2y+{y}^{3}& \text{}\\ {f}_{xy}\left(x,y\right)& =& -2+3{y}^{2}& \text{}\end{array}$ c) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 6x& \text{}\\ {f}_{yy}\left(x,y\right)& =& 6+6xy& \text{}\\ {f}_{xy}\left(x,y\right)& =& -2+3{y}^{2}& \text{}\end{array}$ d) $\begin{array}{rcll}{f}_{xx}\left(x,y\right)& =& 6+6xy& \text{}\\ {f}_{yy}\left(x,y\right)& =& 6x& \text{}\\ {f}_{xy}\left(x,y\right)& =& -2+3{y}^{2}& \text{}\end{array}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
${f}_{x}\left(x,y\right)=3{x}^{2}-2y+{y}^{3},{f}_{y}\left(x,y\right)=-2x+3x{y}^{2}+6y$,
${f}_{xx}\left(x,y\right)=6x,{f}_{yy}\left(x,y\right)=6xy+6$,
${f}_{xy}\left(x,y\right)=-2+3{y}^{2}$.
Choice (d) is incorrect
Find the second order partial derivative with respect to $x$ of $f\left(x,y\right)=cosx+xy{e}^{xy}+xsiny$.
 a) ${f}_{xx}\left(x,y\right)=-cosx+2{y}^{2}{e}^{xy}+x{y}^{3}{e}^{xy}+siny$ b) ${f}_{xx}\left(x,y\right)=2{y}^{2}{e}^{xy}+x{y}^{3}{e}^{xy}$ c) ${f}_{xx}\left(x,y\right)={e}^{xy}+2{y}^{2}{e}^{xy}+x{y}^{3}{e}^{xy}-cosy$ d) ${f}_{xx}\left(x,y\right)=-cosx+2{y}^{2}{e}^{xy}+x{y}^{3}{e}^{xy}$ e) ${f}_{xx}\left(x,y\right)=2{x}^{2}{e}^{xy}+{x}^{3}y{e}^{xy}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Find the second order partial derivative with respect to $y$ of $f\left(x,y\right)=cosx+xy{e}^{xy}+xsiny$.
 a) ${f}_{yy}\left(x,y\right)=2{x}^{2}{e}^{xy}+{x}^{3}y{e}^{xy}-xsiny$ b) ${f}_{yy}\left(x,y\right)=-cosx+2{y}^{2}{e}^{xy}+{x}^{2}{y}^{2}{e}^{xy}+cosy$ c) ${f}_{yy}\left(x,y\right)={e}^{xy}+3xy{e}^{xy}+{x}^{2}{y}^{2}{e}^{xy}+cosy$ d) ${f}_{yy}\left(x,y\right)={x}^{2}{e}^{xy}+{x}^{3}y{e}^{xy}-xsiny$

Choice (a) is correct!
$\begin{array}{rcll}{f}_{y}\left(x,y\right)& =& x{e}^{xy}+{x}^{2}y{e}^{xy}+xcosy& \text{}\\ {f}_{yy}\left(x,y\right)& =& {x}^{3}y{e}^{xy}+{x}^{2}{e}^{xy}+{x}^{2}{e}^{xy}-xsiny& \text{}\\ & =& 2{x}^{2}{e}^{xy}+{x}^{3}y{e}^{xy}-xsiny.& \text{}\end{array}$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
What is the mixed, second order partial derivative of
$f\left(x,y\right)=\frac{3x}{x+y}?$
 a) ${f}_{xx}\left(x,y\right)=\frac{-6y}{{\left(x+y\right)}^{3}}$ b) fyy(x,y) = 6x (x+y)3 c) ${f}_{xy}\left(x,y\right)=\frac{3\left(y-x\right)}{{\left(x+y\right)}^{3}}$ d) ${f}_{xy}\left(x,y\right)=\frac{6\left(x-y\right)}{{\left(x+y\right)}^{3}}$ e) ${f}_{xy}\left(x,y\right)=\frac{3\left(x-y\right)}{{\left(x+y\right)}^{3}}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
${f}_{x}=\frac{3y}{{\left(x+y\right)}^{2}}$, ${f}_{xy}=\frac{3{\left(x+y\right)}^{2}-6y\left(x+y\right)}{{\left(x+y\right)}^{4}}=\frac{3\left(x-y\right)}{{\left(x+y\right)}^{3}}.$
If $f\left(x,y\right)=sin\left(\frac{y}{x}\right)$ which of the following are true ? (More than one answer may be correct.)
 a) $\frac{{\partial }^{2}f}{\partial {x}^{2}}=\frac{2y}{{x}^{3}}cos\left(\frac{y}{x}\right)$ b) $\frac{{\partial }^{2}f}{\partial {y}^{2}}=\frac{-1}{{x}^{2}}sin\left(\frac{y}{x}\right)$ c) $\frac{{\partial }^{2}f}{\partial x\partial y}$ is defined for all $x$ and $y$ d) $\frac{{\partial }^{2}f}{\partial {x}^{2}}=\frac{2}{{x}^{3}}cos\left(\frac{y}{x}\right)-\frac{1}{{x}^{4}}sin\left(\frac{y}{x}\right)$ e) $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$ provided $x\ne 0$ f) $\frac{{\partial }^{2}f}{\partial {x}^{2}}=\frac{2y}{{x}^{3}}cos\left(\frac{y}{x}\right)-\frac{{y}^{2}}{{x}^{4}}sin\left(\frac{y}{x}\right)$

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be True.
There is at least one mistake.
For example, choice (c) should be False.
$f\left(x,y\right)$ itself is not defined when $x=0$. So $\frac{{\partial }^{2}f}{\partial x\partial y}$ is certainly not defined when $x=0$.
There is at least one mistake.
For example, choice (d) should be False.
There is at least one mistake.
For example, choice (e) should be True.
By lectures, $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$ if both mixed derivatives exist and are continuous.
There is at least one mistake.
For example, choice (f) should be True.
Correct!
1. False
2. True
3. False $f\left(x,y\right)$ itself is not defined when $x=0$. So $\frac{{\partial }^{2}f}{\partial x\partial y}$ is certainly not defined when $x=0$.
4. False
5. True By lectures, $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$ if both mixed derivatives exist and are continuous.
6. True