Finding Partial Derivatives with Implicit Differentiation

[ktobrien]

Homework Statement

Use implicit differentiation to find ∂z/∂x and ∂z/∂y
yz = ln(x + z)

The Attempt at a Solution

I came up with
(x+2)/(x+2)(1-xy-yz)

Could someone please help me solve this. I know to treat y as a constant and to multiply all the derivatives of z by ∂z/∂x

 

[rock.freak667]

if y=constant

the left side is just y(∂z/∂x)


so now for ln(x+z) what happens when you differentiate this w.r.t to x ?

 

[ktobrien]

well I get 1/(x+z)(1+(∂z/∂x)). But this is what I did when I got the incorrect answer.

 

[Dick]

That's fine for the right side if you are doing d/dx. What's the left side? Isn't it (dy/dx)*z+y*(dz/dx)? I don't understand how your answer doesn't contain any derivatives.

 

[ktobrien]

Because that's what I am trying find. You differentiate z with respect to x.
y(∂z/∂x)=1/(x+z)(1+(∂z/∂x)) is what I got but I don't think its right and if it is I messed something up when I solved for (∂z/∂x)

 

[Dick]

You can't eliminate all of the derivatives from the solution of either dz/dx or dz/dy. Each solution has to contain the partial derivative of z wrt to the other variable.

 

[ktobrien]

there are two different answers. the answer i got was just for ∂z/∂x. can someone please tell me if I did it right or not.

 

[Dick]

No. You didn't do it right. If you are solving for dz/dx how can you get rid of dy/dx?

 

[ktobrien]

y is a constant when you solve implicitly for (∂z/∂x)

 

[Dick]

y is a constant when you solve implicitly for (∂z/∂x)

Of course it is. Sorry. I wasn't thinking. y(∂z/∂x)=(1/(x+z))*(1+(∂z/∂x)) is fine for a start. Now what do you get when you solve for ∂z/∂x?

 

[ktobrien]

I figured it out. Thanks though. I got the wrong answer because I did the algebra wrong. I just assumed I did the calculus wrong. Thanks again.