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Question
Differentiate \[{10}^\left( {10}^x \right)\] ?
Solution
\[\text{ Let y } = {10}^\left( {10}^x \right) . . . \left( i \right)\]
\[\text{ Taking log on both sides }, \]
\[\log y = \log_e {10}^\left( 10x \right) \]
\[\log y = {10}^x \log_e 10 \]
\[\text{Differentiating with respect to x}, \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \log_e 10 \times {10}^x \log_e 10 \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = {10}^x \times \left( \log_e 10 \right)^2 \]
\[ \Rightarrow \frac{dy}{dx} = y\left[ {10}^x \times \left( \log_e 10 \right)^2 \right] \]
\[ \therefore \frac{dy}{dx} = {10}^\left( 10x \right) \times {10}^x \times \left( \log_e 10 \right)^2 \left[ \text{ using equation } \left( i \right) \right]\]
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