Differentiate the following functions with respect to x :

Let $y=10^{\log \sin x}$

Taking log both the sides:

$\Rightarrow \log y=\log 10^{\log \sin x}$

$\Rightarrow \log y=\log \sin x \log 10\left\{\log x^{a}=\operatorname{alog} x\right\}$

Differentiating with respect to $\mathrm{x}$ :\

$\Rightarrow \frac{d(\log y)}{d x}=\frac{d(\log 10 \log \sin x)}{d x}$

$\Rightarrow \frac{d(\log y)}{d x}=\log 10 \times \frac{d(\log \sin x)}{d x}$

$\left\{\right.$ Using chain rule, $\frac{\mathrm{d}(\mathrm{au})}{\mathrm{dx}}=\mathrm{a} \frac{\mathrm{du}}{\mathrm{dx}}$ where a is any constant and $\mathrm{u}$ is any variable $\}$

$\Rightarrow \frac{1}{y} \frac{d y}{d x}=\log 10 \times \frac{1}{\sin x} \frac{d(\sin x)}{d x}$

$\left\{\frac{d(\log u)}{d x}=\frac{1}{u} \frac{d u}{d x} ; \frac{d(\sin x)}{d x}=\cos x\right\}$

$\Rightarrow \frac{1}{y} \frac{d y}{d x}=\frac{\log 10}{\sin x}(\cos x)$

$\Rightarrow \frac{d y}{d x}=y\{\log 10 \cot x\}$

Put the value of $y=10^{\log \sin x}$ :

$\Rightarrow \frac{d y}{d x}=10^{\log \sin x}\{\log 10 \cot x\}$