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Question
\[\frac{{10}^x}{\sin x}\]
Solution
\[\text{ Let } u = {10}^x ; v = \sin x\]
\[\text{ Then }, u' = {10}^x \log 10; v' = \cos x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{{10}^x}{\sin x} \right) = \frac{\sin x {10}^x \log 10 - {10}^x \cos x}{\sin^2 x}\]
\[ = \frac{\sin x {10}^x \log 10}{\sin^2 x} - \frac{{10}^x \cos x}{\sin^2 x}\]
\[ = {10}^x \log 10 \cos ec x - {10}^x cosec x \cot x\]
\[ = {10}^x cosec x\left( \log 10 - \cot x \right)\]
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