Question

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 y=

part 6)

Find the following partial derivatives of the function f(x,y,z)=x^8+2yz^2+z^5

fx(x,y,z)=

fy(x,y,z)=

fz(x,y,z)=

fyz(x,y,z)=

Answer #1

Find all second partial derivatives of f : ?(?, ?, ?) =
?^??^??^?
fx=
fy=
fz=
fxx=
fyy=
fzz=
fxy=
fxz=
fyz=

(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=

Please find ALL second partial derivatives of f: fx, fy, fz,
fxx, fyy, fzz, fxy, fxz, and fyz
For ?(?, ?, ?) = (?^?)(?^?)(?^?)
THANK YOU

Let f(x, y) = 2x^3y^2 + 3xy^3 4x^3 y. Find
(a) fx
(c) fxx
(b) fy
(d) fyy
(e) fxy
(f) fyx

Let
f(x,y,z)=exy4z5+xy2+y3z
Calculate.
fx fy fz fxx fxy fyy
fxz fyz fzz fzyy fxxy fxxyz
1

Consider the function f(x,y) = xe^((x^2)-(y^2))
(a) Find f(1,−1), fx(1,−1), fy(1,−1). Use these values to find a
linear approximation for f (1.1, −0.9).
(b) Find fxx(1, −1), fxy(1, −1), fyy(1, −1). Use these values to
find a quadratic approximation for f(1.1,−0.9).

Suppose that f is a twice differentiable function and that
its second partial derivatives are continuous. Let h(t) =
f (x(t), y(t)) where x = 2e^ t and y = 2t. Suppose that
fx(2, 0) = 1, fy(2, 0) = 3, fxx(2, 0) = 4, fyy(2, 0) = 1, and
fxy(2, 0) = 4. Find d ^2h/ dt ^2 when t = 0.

Suppose that f is a twice differentiable function and that
its second partial derivatives are continuous. Let h(t) =
f (x(t), y(t)) where x = 3e ^t and y = 2t. Suppose that
fx(3, 0) = 2, fy(3, 0) = 1, fxx(3, 0) = 3, fyy(3, 0) = 2, and
fxy(3, 0) = 1. Find d 2h dt 2 when t = 0.

. For the function x,y=xarctan(xy) , compute
fx , fy ,
fxx , fyy , and
fxy

Suppose that the function f(x, y) has continuous partial
derivatives fxx, fyy, and fxy at all points (x,y) near a critical
points (a, b). Let D(x,y) = fxx(x, y)fyy(x,y) – (fxy(x,y))2 and
suppose that D(a,b) > 0.
(a) Show that fxx(a,b) < 0 if and only if fyy(a,b) <
0.
(b) Show that fxx(a,b) > 0 if and only if fyy(a,b) >
0.

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