I was wondering whether anybody could explain how you derive this implicit differentiation rule: (del z)/(del x)=−(del f/ del x)/(del f/ del z) if you have a function z implicity defined by f(x,y,z)=0?

Lorelei Patterson

Lorelei Patterson

Answered question

2022-07-22

I was wondering whether anybody could explain how you derive this implicit differentiation rule:
z x = f / x f / z
if you have a function z implicity defined by f ( x , y , z ) = 0?

Answer & Explanation

abortargy

abortargy

Beginner2022-07-23Added 19 answers

Take z a function of x , y. Then it is always true that
f ( x , y , z ( x , y ) ) = 0
Its derivative is also 0. So
f x d x + f x d y + f z ( z x d x + f x d y ) = 0.
If we consider the case d y = 0, d y 0, then divide by d x and cross out d y,
f x + f z z x = 0.
Or,
z x = f / x f / z

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?