# Multi-Variable Calc: implicit/chain differentiation

Find partial derivative of $z$ with respect to the partial of $y$ using the result of the chain rule.

$$\ln(x^2+y^2) + x\ln(z) - \cos(xyz)=3.$$

I would use regular implicit differentiation for this problem but what does "using result of the chain rule" mean?

Any help would be appreciated. :)

• You might want to clarify which variables are independent and which are dependent. – copper.hat Apr 10 '17 at 2:57
• Clarification: z is a function of x and y – idk Apr 10 '17 at 3:35

$$\frac{\partial}{\partial_y} (f(x,y,z(x,y))) = 0 \implies \frac{\partial f}{\partial y} (x,y,z(x,y)) + \frac{\partial z}{\partial y}(x,y) \frac{\partial f}{\partial z}(x,y, z(x,y)) = 0$$ $$\implies \frac{\partial z}{\partial y} = - \frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial z}}$$