Question

Find dy/dx, using implicit differentiation.
4x + 7y = xy
dy
dx=
Compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative.
y=
dy
dx=
I am stuck on this problem

Answer 1

If you take dy/dx, you get;
4(dx/dx)+7(dy/dx)=d(xy)/dx
4+7(dy/dx)=x(dy/dx)+y(dx/dx)
4+7(dy/dx)=x(dy/dx)+y
dy/dx(7-x)=y-4
dy/dx=(y-4)/(7-x)
Just find in the original equation, y in terms of x.
Now, can you do it?

Answer 2

one second

Answer 3

4(dx/dx) that first dx is dy correct?

Answer 4

No, we are finding the derivate of the whole thing with respect to x. So, whatever there is in the numerator, we will differentiate it with respect to x. (dx/dx) will be 1

Answer 5

oh okay

Answer 6

okay so fir this part x(dy/dx)+y(dx/dx) why is the y(dx/dx) not y(dy/dx)?

Answer 7

4x + 7y = xy
y': 4 + 7y' = y + xy'
x': 4x' + 7 = x'y + x

Answer 8

In plain version :)

Answer 9

y': 4 + 7y' = y + xy' that would be the only one i need correct instead of the x'?

Answer 10

Compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative.
y =

Answer 11

how would i do that step now?

Answer 12

What did you come up?

Answer 13

y = 4 + 7y/x

Answer 14

Solving for y as function of x

Answer 15

4x/x-7 is what i got and it was correct

Answer 16

but i have no idea how to get that answer...

Answer 17

For the second part?

Answer 18

yeah

Answer 19

Let me do on paper first!

Answer 20

dy/dx=(y-4)/(7-x)
this was the answer to the first part

Answer 21

okay

Answer 22

I don't see where to get the denominator ( 7-x)?

Answer 23

y': 4 + 7y' = y + xy'
7y' - xy' = y -4
->( 7 -x) y' = y -4
=> y' = ( y - 4)/ ( 7 -x)

Answer 24

4x+7y=xy is the original equation
so you move the xy to the side with 7y
xy-7y=4x
then you factor out a y
y(x-7)=4x
then you divide by x-7
y=4x/(x-7)

Answer 25

I forgot we have to factor it out y' first :)

Answer 26

If you take derivative of y = 4x/(x-7)
I can't find the match with the result of first answer :(

Answer 27

@Mani Jha, give some clue for the second part will you!

Answer 28

y = 4 + 7y/x ??

Answer 29

The second part should give the same answer as the first. Wait a second

Answer 30

Oh, I got it

Answer 31

I guess because I'm sort of relyng on you

Answer 32

y = 4x/ ( 7 -x)
y' = 4 (7-x) - 4x/ ( 7-x)^2

Answer 33

That should be x-7 downstairs, isnt it?

Answer 34

= 4 ( 7-x)/ (7-x)^2

Answer 35

Cancel out

Answer 36

y=4x/(x-7)
y'={(x-7)4-4x}/(x-7)^2
=-28/(x-7)^2
Now we substitute y=4x/(x-7) in our first result
we get -28/(x-7)^2, which is same
I still dont see how you get 7-x downstairs!

Answer 37

(28 - 8x)/ (7-x)^2

Answer 38

you substituted y=4x/(x-7), right?
y' = {(4x)/(x-7)-4}/(7-x)
={(4x-4x+28)/(7-x)(x-7)}
=28/(7-x)(x-7)
=-28/(x-7)^2

Answer 39

OH, that's my problem. All I do is take derivative!

Answer 40

@camaron, do you understand what Mani has showed?
@Mani, thanks a lot! I learn the new substitution today :)

Answer 41

No problem, dude ;)