For the functions w=xy+yz+xz, x=u+2v, y=u-2v, and z=uv, express ζw/ζu and ζw/ζv using the chain rule and by expressing w directly in terms of u and v before differentiating. Then evaluate ζw/ζu and ζw/ζv at the point (u, v)=(2/3,0)

i) See explanations below

ii) partial derivative of w with respect to u = 2u + 4uv

partial derivative of w with respect to v = -4v + 2u²

The partial derivative of w with respect to u = 4/3

The partial derivative of w with respect to v = 8/9

Step-by-step explanation:

w=xy+yz+xz

x=u+2v

y=u-2v

z=uv

First we are to express partial derivative of w with respect to u and v respectively using chain rule.

See 1st attachment.

Inserting the value of x, y and z with respect to u and v in w:

w = (u+2v)(u-2v) + (u-2v)(uv) + (u+2v)(uv)

w = u² -2uv +2uv - 2v² +u²v -2uv² + u²v + 2uv²

w = u² - 2v² + 2u²v

We would find the partial derivative of w with respect to u and v respectively.

Partial derivative is done by differentiating a function with one variable while keeping the other variable constant.

At the point (2/3, 0) given, we get the value for the partial derivative of w with respect to u and v respectively.

See 2nd attachment for more explanations.

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕