Incorrect Implicit differentiation answer

[craggy scarab]

I'm really confused I thought this was an implicit deafferentation question

But the answer (2) I assume implies it was just a partial derivative question where the answer will be 2.

How can I better spot the differences between these 2 types of questions in the future (Unless I just messed up my calculation for the implicit differentiation calculation.

I just completed this exam got home and they provided the test with solved answers (a) I am trying to go over my mistakes and I just cant seem to figure this one out could anyone help?

[languid kestrel]

Incorrect Implicit differentiation answer

[atomic tiger]

your answer is correct for the partial derivative

however if you plug (1,1,2) into the original equation 1x1x2-1x2^2 = -2 which is not equal to 0 so it doesnt pass through the point so doesnt have a derivative at the point as it is undefined

[gusty token]

This requires the implicit function theorem

If F(x,y,z) = xyz - z^2 = 0, then we let x = x(u) = u, y = y(v) = v, z = z(u, v).

-> F(x,y,z) = F(x(u), y(v), z(u,v)) = 0
By chain rule:
-> F_x = F_x * partial_x/partial_u + F_y * partial_y/partial_u + F_z * partial_z/partial_u

Latex may be nicer for this

[forest crypt]

This has already been asked btw

the task is flawed

(1,1,2) doesn't satisfy the equation

just so you know

[gusty token]

Ffs, started writing a long latex scripture without checking

[craggy scarab]

😂 I appreciate you double checking for me man

This was for an exam I just took today aswell

So I guess the teacher either got it wrong or theres something im missing I guess