Differentiate the following functions with respect to x: 10^ (10^… - Vidyadip

Question

Differentiate the following functions with respect to x:
$10^{(10^\text{x})}$

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Answer

Let $\text{y}=10^{(10^\text{x})}\ .....(\text{i})$
Taking log on both sides,
$\log\text{y}=\log_\text{e}10^{(10^\text{x})}$
$\log\text{y}=10^{\text{x}}\log_\text{e}10$
Differentiating with respect to x,
$\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log_\text{e}10\times10^\text{x}\log_\text{e}10$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=10^\text{x}\times(\log_\text{e}10)^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\big[10^{\text{x}}\times(\log_\text{e}10)^2\big]$
$\therefore\ \frac{\text{dy}}{\text{dx}}=10^{(10\text{x})}\times10^\text{x}\times(\log_\text{e}10)^2$
[Using equation (i)]

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