Question

Assume that all the given functions have continuous second-order partial derivatives. If z = f(x, y), where x = r2 + s2 and y = 6rs, find ∂2z/∂r∂s. (Compare with Example 7.) ∂2z/∂r∂s = ∂2z/∂x2 + ∂2z/∂y2 + ∂2z/∂x∂y + ∂z/∂y

Answer #1

2. (a)
Determine all first and second order partial derivatives of
f(x,y,z) = x2y3 sin(xz)
(b) Determine all first-order partial derivatives of
g(x,y,z)=u2y+x2v where u=exz,
v=sin(yz)

(1 point)
Find all the first and second order partial derivatives of
f(x,y)=7sin(2x+y)−2cos(x−y)
A. ∂f∂x=fx=∂f∂x=fx=
B. ∂f∂y=fy=∂f∂y=fy=
C. ∂2f∂x2=fxx=∂2f∂x2=fxx=
D. ∂2f∂y2=fyy=∂2f∂y2=fyy=
E. ∂2f∂x∂y=fyx=∂2f∂x∂y=fyx=
F. ∂2f∂y∂x=fxy=∂2f∂y∂x=fxy=

Find the first- and second-order partial derivatives for the
following function. z = f (x, y) = (ex +1)ln y.

Find all the second order partial derivatives of the
function
w = x - y / x2 + y

Given w=f(x,y,z)
List all of the second and third derivatives. How many unique
second derivatives? How many unique third derivatives?
Example: If z=f(x,y) , then z has 3 unique derivatives.
fxx, fxy. fyy

Compute all second order partial derivatives of f(x, y) = x^4 −
2x ^3 y^2

Calculate all four second-order partial derivatives for the
function f(x,y)=3+6x2y2 ?

find all the second partial derivatives of f(x,y)=
x^2y^2+3sinxtany

Find all second partial derivatives of the function:
f(x,y) = x2y - 3xy3

part 1)
Find the partial derivatives of the function
f(x,y)=xsin(7x^6y):
fx(x,y)=
fy(x,y)=
part 2)
Find the partial derivatives of the function
f(x,y)=x^6y^6/x^2+y^2
fx(x,y)=
fy(x,y)=
part 3)
Find all first- and second-order partial derivatives of the
function f(x,y)=2x^2y^2−2x^2+5y
fx(x,y)=
fy(x,y)=
fxx(x,y)=
fxy(x,y)=
fyy(x,y)=
part 4)
Find all first- and second-order partial derivatives of the
function f(x,y)=9ye^(3x)
fx(x,y)=
fy(x,y)=
fxx(x,y)=
fxy(x,y)=
fyy(x,y)=
part 5)
For the function given below, find the numbers (x,y) such that
fx(x,y)=0 and fy(x,y)=0
f(x,y)=6x^2+23y^2+23xy+4x−2
Answer: x= and...

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