# Write first and second partial derivatives f(x,y)=2xy+9x^2y^3+7e^(2y)+16 a)f_x b)f_(xx) c)f_(xy) d)f_y e)f_(yy) f)f_(yx)

Question
Multivariable functions
Write first and second partial derivatives
$$\displaystyle{f{{\left({x},{y}\right)}}}={2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{7}{e}^{{{2}{y}}}+{16}$$
a)$$\displaystyle{f}_{{x}}$$
b)$$\displaystyle{f}_{{\times}}$$
c)$$\displaystyle{f}_{{{x}{y}}}$$
d)$$\displaystyle{f}_{{y}}$$
e)$$\displaystyle{f}_{{{y}{y}}}$$
f)$$\displaystyle{f}_{{{y}{x}}}$$

2021-02-03
a) $$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{7}{e}^{{{2}{y}}}+{16}\right]}$$
$$\displaystyle{f}_{{x}}={2}{y}+{18}{x}{y}^{{3}}+{0}+{0}={2}{y}+{18}{x}{y}^{{3}}$$
$$\displaystyle\Rightarrow{f}_{{x}}={2}{y}+{18}{x}{y}^{{3}}$$
b) $$\displaystyle{f}_{{\times}}=\frac{{\partial{f}_{{x}}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{9}{y}+{18}{x}{y}^{{3}}\right]}$$
$$\displaystyle={0}+{18}{y}^{{3}}={18}{y}^{{3}}$$
$$\displaystyle{f}_{{\times}}={18}{y}^{{3}}$$
с) $$\displaystyle{f}_{{{x}{y}}}=\frac{{\partial{f}_{{x}}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{2}{y}+{18}{x}{y}^{{3}}\right]}$$
$$\displaystyle{f}_{{{x}{y}}}={2}+{18}{x}\cdot{3}{y}^{{2}}$$
$$\displaystyle{f}_{{{x}{y}}}={2}+{54}{x}{6}^{{2}}$$
d) $$\displaystyle{f}_{{y}}=\frac{{\partial{f}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{4}{e}^{{{2}{y}}}+{16}\right]}$$
$$\displaystyle{f}_{{y}}={2}{x}+{9}{x}^{{2}}\cdot{3}{y}^{{2}}+{7}\cdot{2}{e}^{{{2}{y}}}+{0}$$
$$\displaystyle{f}_{{y}}={9}{x}+{27}{x}^{{2}}{y}+{14}{e}^{{2}}{y}$$
e) $$\displaystyle{f}_{{{y}{y}}}=\frac{\partial}{{\partial{y}}}{\left[{2}{x}+{27}{x}^{{2}}{y}^{{2}}+{14}{e}^{{{2}{y}}}\right.}$$
$$\displaystyle={0}+{27}{x}^{{2}}{x}^{{y}}^{2}+{14}\cdot{2}{e}^{{{2}{y}}}$$
$$\displaystyle{f}_{{y}}{y}={54}{x}^{{2}}\cdot{2}{y}+{28}{e}^{{{2}{y}}}$$
f) $$\displaystyle{f}_{{{y}{x}}}=\frac{{\partial{f}_{{y}}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{2}{x}+{27}{x}^{{y}}^{2}+{14}{e}^{{{2}{y}}}\right]}$$
$$\displaystyle{f}_{{y}}{y}={2}={54}{x}{y}^{{2}}$$

### Relevant Questions

A surface is represented by the following multivariable function,
$$\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{3}}{x}^{{3}}+{y}^{{2}}-{2}{x}{y}-{6}{x}-{3}{y}+{4}$$
a) Calculate $$\displaystyle{f}_{{\times}},{f}_{{{y}{x}}},{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{y}}}$$
b) Calculate coordinates of stationary points.
c) Classify all stationary points.
Write first and second partial derivatives
$$\displaystyle{g{{\left({r},{t}\right)}}}={t}{\ln{{r}}}+{11}{r}{t}^{{7}}-{5}{\left({8}^{{r}}\right)}-{t}{r}$$
a)$$\displaystyle{g}_{{r}}$$
b)$$\displaystyle{g}_{{{r}{r}}}$$
c)$$\displaystyle{g}_{{{r}{t}}}$$
d)$$\displaystyle{g}_{{t}}$$
e)$$\displaystyle{g}_{{{\mathtt}}}$$
Consider this multivariable function. $$\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{{3}{x}}}+{y}^{{2}}$$
a) Find $$\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}$$
b) What is value of $$\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}$$?
COnsider the multivariable function $$\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{2}}-{3}{y}^{{4}}{x}^{{2}}+{\sin{{\left({x}{y}\right)}}}$$. Find the following partial derivatives: $$\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}$$.
Write formulas for the indicated partial derivatives for the multivariable function.
$$\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}$$
a)$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}$$
b)(delf)/(dely)ZSK
c)$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}$$
Write formulas for the indicated partial derivatives for the multivariable function. $$\displaystyle{g{{\left({x},{y},{z}\right)}}}={3.1}{x}^{{2}}{y}{z}^{{2}}+{2.7}{x}^{{y}}+{z}$$
a)$$\displaystyle{g}_{{x}}$$
b)$$\displaystyle{g}_{{y}}$$
c)$$\displaystyle{g}_{{z}}$$
a) Find the function's domain .
b) Find the function's range.
c) Find the boundary of the function's domain.
d) Determine if the domain is an open region, a closed region , or neither.
e) Decide if the domain is bounded or unbounded.
for: $$\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{{\left({16}-{x}^{{2}}-{y}^{{2}}\right)}^{{\frac{{1}}{{2}}}}}}$$
Write formulas for the indicated partial derivatives for the multivariable function.
$$\displaystyle{k}{\left({a},{b}\right)}={3}{a}{b}^{{4}}+{8}{\left({1.4}^{{b}}\right)}$$
a) $$\displaystyle\frac{{\partial{k}}}{{\partial{a}}}$$
b) $$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}$$
c) $$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}$$
Mixed Partial Derivatives If f is a function of a and y such that $$\displaystyle{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{x}}}$$ are continuous, what is the relationship between the mixed partial derivatives?
$$\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{4}}{m}^{{5}}−{3}{k}{m}$$
a)$$\displaystyle{g}_{{k}}$$
b)$$\displaystyle{g}_{{m}}$$
c)$$\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}$$