3.7 - Implicit Functions: find dy/dx at xy + x + y = 5

T

tglass

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#3. xy+x+y=5

So, dy/dx(xy)+dy/dx(x)+dy/dx(y) = dy/dx(5)

dy/dx(xy) -> we must use the product rule. So (x)'*(y)+(x)*dy/dx(y). (x)' = 1, so y+x*dy/dx(y).
dy/dx(5) = 0

So:
y+x*dy/dx(y)+1+dy/dx(y) = 0

y+x*dy/dx(y)+dy/dx(y) = -1

x*dy/dx(y)+dy/dx(y) = -1-y
Factor out the dy/dx(y)
dy/dx(y)*(x+1)=-1-y
Divide by x+1
dy/dx(y) = -1-y/(x+1)
Divide by y
dy/dx = [(-1-y)*y]/(x+1)

However, this is not the answer the book has. Am I wrong in assuming that I must use the product rule on the xy factor? Or have I made some error somewhere else?

Thanks in advance as always,
TG
 
Re: 3.7 - Implicit Functions

Why are you dividing by y at the end? I get \(\displaystyle y' = \frac{-(y+1)}{x+1}\).
 
Re: 3.7 - Implicit Functions

I thought I had to to get dy/dx completey by itself.

From an example in the book:
x^2+y^2=4

So dy/dx(x^2)+dy/dx(y^2)=dy/dx(4)

2x+dy/dx(2y)=0

dy/dx(2y)=-2x

dy/dx=(-2x)/(2y) = -x/y

They divided the 2y term off of the dy/dx to get alone, why wouldn't I do similarly with the problem I've given?
 
tglass said:
#3. xy+x+y=5

So, dy/dx(xy)+dy/dx(x)+dy/dx(y) = dy/dx(5)

dy/dx(xy) -> we must use the product rule. So (x)'*(y)+(x)*dy/dx(y). (x)' = 1, so y+x*dy/dx(y).
dy/dx(5) = 0

So:
y+x*dy/dx(y)+1+dy/dx(y) = 0

y+x*dy/dx(y)+dy/dx(y) = -1

x*dy/dx(y)+dy/dx(y) = -1-y
Factor out the dy/dx(y)
dy/dx(y)*(x+1)=-1-y
Divide by x+1
dy/dx(y) = -1-y/(x+1)
Divide by y
dy/dx = [(-1-y)*y]/(x+1)

However, this is not the answer the book has. Am I wrong in assuming that I must use the product rule on the xy factor? Or have I made some error somewhere else?

Thanks in advance as always,
TG
Click to expand...
....................................
This line:

dy/dx(xy)+dy/dx(x)+dy/dx(y) = dy/dx(5)

is wrong. It should say:

d/dx(xy)+d/dx(x)+d/dx(y) = d/dx(5)

That is how you seem to have an extra 'y' in there.

You should have next:

x dy/dx + y + 1 + dy/dx = 0

Then you'll be ok.
 
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