A) \[y\left[ {{x}^{x}}(logex)logx+{{x}^{x}} \right]\]
B) \[y\left[ {{x}^{x}}(logex)logx+x \right]\]
C) \[y\left[ {{x}^{x}}(logex)logx+{{x}^{x-1}} \right]\]
D) \[y\left[ {{x}^{x}}(lo{{g}_{e}}x)logx+{{x}^{x-1}} \right]\]
Correct Answer: C
Solution :
[c] \[y={{x}^{({{x}^{x}})}}\] Or \[\log y={{x}^{x}}\log x\] Or \[\frac{1}{y}\frac{dy}{dx}=\frac{dz}{dx}\log x+\frac{1}{x}z\] (where\[{{x}^{x}}=z\]) Or \[\frac{dy}{dx}={{x}^{({{x}^{x}})}}\left[ {{x}^{x}}(lo{{g}_{e}}x)logx+{{x}^{x-1}} \right]\]\[\left( \therefore \frac{dz}{dx}={{x}^{x}}{{\log }_{e}}x \right)\]
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