Implicit partial differentiation

[morsel]

Homework Statement

Find \partial x / \partial z at the point (1, -1, -3) if the equation xz + y \ln x - x^2 + 4 = 0 defines x as a function of the two independent variables y and z and the partial derivative exists.

Homework Equations

The Attempt at a Solution

x + y/x \partial x / \partial z - 2x \partial x / \partial z = 0

Did I do the implicit differentiation correctly? I'm unsure about where to put \partial x / \partial z when I differentiate.

Thanks!

 

[Mark44]

Homework Statement

Find \partial x / \partial z at the point (1, -1, -3) if the equation xz + y \ln x - x^2 + 4 = 0 defines x as a function of the two independent variables y and z and the partial derivative exists.

Homework Equations

The Attempt at a Solution

x + y/x \partial x / \partial z - 2x \partial x / \partial z = 0

Did I do the implicit differentiation correctly? I'm unsure about where to put \partial x / \partial z when I differentiate.

No, this isn't correct. When you take the partial of y lnx with respect to z you have to use the product rule (and then the chain rule), and as far as I can tell, you didn't use it.

Once you have differentiated both sides of the equation, solve algebraically for \frac{\partial x}{\partial z}

 

[betel]

I think the ln term is correct. But the first term is not. Here you have to use the product and chain rule.

 

[Mark44]

I think the ln term is correct. But the first term is not. Here you have to use the product and chain rule.

I didn't check, but I think you're probably right. Having x be the dependent variable and y and z independent probably threw me off.