A) \[\frac{1+\log x}{1+\log z}\]
B) \[-\frac{1+\log x}{1+\log z}\]
C) \[-\frac{1+\log y}{1+\log z}\]
D) None of these
Correct Answer: B
Solution :
\[{{x}^{x}}{{y}^{y}}{{z}^{z}}=c\]Þ \[\log ({{x}^{x}}{{y}^{y}}{{z}^{z}})=\log c\] Þ \[x\log x+y\log y+z\log z=\log c\] .....(i) Here x, y are regarded as independent variables and z depends on x, y. Differentiating (i) partially w.r.t. ?x? \[x.\frac{1}{x}+\log x.1+0+\left( z.\frac{1}{z}+\log z.1 \right)\frac{\partial z}{\partial x}=0\] \[\therefore \] \[\frac{\partial z}{\partial x}=-\frac{1+\log x}{1+\log z}\].
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