Use implicit differentiation for the function: yz = ln(x+z) to find the partial derivative of both y and z in respect to x

Question

anonymous · Answer

Any ideas on how to start this question?

irishboy123 · Answer

for $\dfrac{\partial z}{\partial x}$ just do normal implicit diff with $z = z(x)$ and $y = const$

vice versa for $\dfrac{\partial y}{\partial x}$

that make sense?!

anonymous · Answer

yeah I got something like:
$$ \frac{ ∂y}{ ∂x }z  = \frac{ 1 }{ x+z }$$

anonymous · Answer

Does it seem about right?
And then from here, I would make ∂y/∂x the subject

irishboy123 · Answer

looks good

irishboy123 · Answer

yeah.  make it the subject

anonymous · Answer

Okay cool, I guess this would also be a similar outcome for ∂z/∂x

irishboy123 · Answer

for z, i get

$ y \dfrac{\partial z}{\partial x} = \dfrac{1}{x+z} (1 + \dfrac{\partial z}{\partial x})$

and then tidy up

anonymous · Answer

Okay cool got it!