Differentiate $\displaystyle \log \left(\mathrm{x}^{\mathrm{x}}+\operatorname{cosec}^{2} \mathrm{x}\right)$ with respect to x .

\[\begin{aligned} \frac{d}{d \mathrm{x}} \log \left(\mathrm{x}^{\mathrm{x}}+\operatorname{cosec}^{2} \mathrm{x}\right) & =\frac{1}{\mathrm{x}^{\mathrm{x}}+\operatorname{cosec}^{2} \mathrm{x}} \frac{d}{d \mathrm{x}}\left(e^{\mathrm{x} \log \mathrm{x}}+\operatorname{cosec}^{2} \mathrm{x}\right) \quad\left(\because \mathrm{x}^{\mathrm{x}}=e^{\mathrm{x} \log \mathrm{x}}\right) \\ & =\frac{1}{\mathrm{x}^{\mathrm{x}}+\operatorname{cosec}^{2} \mathrm{x}}\left[e^{\mathrm{x} \log \mathrm{x}}(1+\log \mathrm{x})-2 \operatorname{cosec}^{2} \mathrm{x} \cot \mathrm{x}\right] \\ & =\frac{1}{\mathrm{x}^{\mathrm{x}}+\operatorname{cosec}^{2} \mathrm{x}}\left[\mathrm{x}^{\mathrm{x}}(1+\log \mathrm{x})-2 \operatorname{cosec}^{2} \mathrm{x} \cot \mathrm{x}\right] \end{aligned} \]