If f'(1) = 2 and y=f( _ex), find d dx at x=e. - Vidyadip

Question

If f'(1) = 2 and $\text{y}=\text{f}(\log_\text{e}\text{x}),$ find $\frac{\text{d}}{\text{dx}}\text{at x}=\text{e}.$

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Answer

We have, f'(1) = 2 and $\text{y}=\text{f}(\log_\text{e}\text{x})$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\text{f}'(\log_\text{e}\text{x})\times\frac{\text{d}}{\text{dx}}(\log_\text{e}\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{f}'(\log_\text{e}\text{x})\big(\frac{1}{\text{x}}\big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{f}'(\log_\text{e}\text{e})\big(\frac{1}{\text{e}}\big) \big[\because\text{x}=\text{e}\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{f}'(1)\big(\frac{1}{\text{e}}\big) \big[\because\log_\text{e}\text{e}=1\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2}{\text{e}} \big[\because\text{f}'(1)=2\big]$

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