When is a simple assumption not so simple?

[ngm01]

Greetings, I faced with a problem that states - Find slope of the tangent line to 〖 x〗^4 –

xy^2+ 4xy^2 = 20,at (1,2)

First I set F= x^4 – xy^2+ 4xy^2-20=0

I found dF/dx = 4x^3 + y^2+ 4y^2= 24 ,at (1,2)

then dF/dy = 2xy+8xy,=20 at (1,2) …. Armed with this I then made the assumption that

dy/dx = dF/dX multiplied dy/dF where dy/dF is simply the inverse of dF/dy which led

to dy/dx = 24/20 = 1.2…. however that’s not the answer. The answer is showed as – 1.2,

explained as the perpendicular slope is 20/24 and the tangent slope as - 24/20

Can you tell me what I’m missing here?

Thanks

 

[HallsofIvy]

Greetings, I faced with a problem that states - Find slope of the tangent line to 〖 x〗^4 –

xy^2+ 4xy^2 = 20,at (1,2)

First I set F= x^4 – xy^2+ 4xy^2-20=0

I found dF/dx = 4x^3 + y^2+ 4y^2= 24 ,at (1,2)

then dF/dy = 2xy+8xy,=20 at (1,2) …. Armed with this I then made the assumption that

First, you have lost the "-" on -xy^2. But even with that, your assumption is not true.
If F(x, y)= 0 (or any constant) then, by the chain rule $F_x+ F_y (dy/dx)= 0$. From that, $F_y(dy/dx)= -F_x$ so $dy/dx= -F_x/F_y$, not $F_x/F_y$.

dy/dx = dF/dX multiplied dy/dF where dy/dF is simply the inverse of dF/dy which led

to dy/dx = 24/20 = 1.2…. however that’s not the answer. The answer is showed as – 1.2,

explained as the perpendicular slope is 20/24 and the tangent slope as - 24/20

Can you tell me what I’m missing here?

Thanks

 

[ngm01]

Thanks!