CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY2 Marks
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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