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| Find the derivative of $(\log x)^{\log x},x>1$ |
$(1+\log(\log x)/x)$ $(\log x)^{\log x}(\frac{1+\log(\log x)}{x})$ $(\log x)^{\log x}$ $(\log x)^{1+\log x}$ |
| $(\log x)^{\log x}(\frac{1+\log(\log x)}{x})$ |
| Let $y=(\log x)^{\log x}$. Taking log in both sides we get $\log y=\log x\log(\log x)$. Differentiating both sides w.r.to x we get $1/y\frac{dy}{dx}=\frac{1+\log(\log x)}{x}$. Hence $\frac{dy}{dx}=y(\frac{1+\log(\log x)}{x})=(\log x)^{\log x}(\frac{1+\log(\log x)}{x})$ |
