Question

For the function *w=f(x,y)* , *x=g(u,v)* , and
*y=h(u,v)*. Use the Chain Rule to

Find *∂w**/∂u* and
*∂w**/∂v* when *u=2* and *v=3* if
*g**(2,3)**=4, h**(2,3**)=-2,*
*g**u**(2,3)**=-5,*

*g**v**(2,3)**=-1* ,
*h**u**(2,3)**=3,*
*h**v**(2,3)**=-5,*
*f**x**(4,-2)**=-4, and*
*f**y**(4,-2)**=7*

*∂w**/∂u**=*

*∂w**/∂v* =

Answer #1

Suppose f is a differentiable function of x
and y, and
g(u, v) =
f(eu
+ sin(v),
eu +
cos(v)).
Use the table of values to calculate
gu(0, 0)
and
gv(0, 0).
f
g
fx
fy
(0, 0)
0
5
1
4
(1, 2)
5
0
6
3
gu(0, 0)
=
gv(0, 0)
=

g (u, v) is a differentiable function and g (1,2) = 100, gu
(1,2) = 3, gv (1,2) = 7 are given. The function f is defined as f
(x, y, z) = g (xyz, x ^ 2, y^2z). Find the equation of the tangent
plane at the point (1,1,1) of the f (x, y, z) = 100 surface.

Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...

Use the Chain Rule to evaluate the partial derivative ∂g/∂u at
the point (u,v)=(0,1), where g(x,y)=x^2−y^2, x=e^3ucos(v),
y=e^3usin(v).
(Use symbolic notation and fractions where needed.)

Use the Chain Rule to evaluate the partial derivative
∂f∂u and ∂f∂u at (u, v)=(−1, −1), where
f(x, y, z)=x10+yz16,
x=u2+v, y=u+v2, z=uv.
(Give your answer as a whole or exact number.)
∂f∂u=
∂f∂v=

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...

1020) y=8/sqrt(4x^7)=Ax^B + C. y'=Kx^F + G. y'(9)=H. Find A,B,C,K,F,G,H. ans:7
1026) Find the derivative of y=(5x^(1/4) - 8x^(1/7))^3, when x=2. ans:1
1010) y=3x^2 + 2x + 5 and y'=Ax^2 + Bx + C. Find A,B,C. Next Question:
y=(x^2)/7 + x/4 + 2 and y'=Hx^2 + Fx + G. Find H,F,G. ans:6

Use the Chain Rule to find the indicated partial
derivatives.
z = x2 + xy3,
x = uv2 + w3,
y = u + vew
when u = 2, v = 2, w = 0

Consider the following function.
f (x, y) = [(y +
2) ln x] − xe7y −
x(y − 5)7
(a)
Find fx(1, 0) .
(b)
Find fy(1, 0) .

Let g(u, v) = f(u 3 − v 3 , v3 − u 3 ). Prove that v^2 ∂g/∂u −
u^2 ∂g/∂v = 0, using the Chain Rule

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