If $x^x y^y z^z=c$, then $\frac{\partial z}{\partial x}$ is equal to

Question

Professor PC Thomas Classes · Answer

Take log: $x\ln x + y\ln y + z\ln z = \ln c$. Diff w.r.t $x$: $1+\ln x + (1+\ln z)\frac{\partial z}{\partial x} = 0 \implies \frac{\partial z}{\partial x} = -\frac{1+\log x}{1+\log z}$.