If $F(x, y, z)=0$ find $ \frac {\delta z}{\delta x} , \frac {\delta z}{\delta y}$

Asked Jul 25 '21 at 18:09

Active Jul 25 '21 at 18:56

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We can assume that x is represented as a function of y and z (since F is an implicit function), so y and z are independent.

to find $\frac {\delta z}{\delta x}$ do we need to show all possibilities like:

case 1: x is a function of y and z

case 2: y is a function of x and z

case 3: z is a function of x and y

Edit:

Consider z as a function of x and y,

$\frac {\delta F}{\delta x}$ = $\frac {\delta F}{\delta x}\frac {\delta x}{\delta x}$ + $\frac {\delta F}{\delta y}\frac {\delta y}{\delta x}$ + $\frac {\delta F}{\delta z}\frac {\delta z}{\delta x}$

Here $\frac {\delta y}{\delta x}$ will be zero since x and y are independent.

So, $\frac {\delta F}{\delta x}$ = $\frac {\delta F}{\delta x}$ + $\frac {\delta F}{\delta z}\frac {\delta z}{\delta x}$

I am stuck here. Any suggestions?

edited Jul 25 '21 at 18:56

asked Jul 25 '21 at 18:09

Anup patel

You are taking the derivative of z with respect to x and with respect to y. So you are assuming that z is a function of x and y. – Aaron Jul 25 '21 at 18:35
Ok got it. Thanks @Aaron – Anup patel Jul 25 '21 at 18:45
@Aaron can you help me in finding dz/dx as i am stuck in expansion part – Anup patel Jul 25 '21 at 18:51
It looks like what you have is right. Now solve algebraically for $\frac{\partial z}{\partial x}$ – Aaron Jul 25 '21 at 19:25

If $F(x, y, z)=0$ find $ \frac {\delta z}{\delta x} , \frac {\delta z}{\delta y}$

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