Find the second order derivatives of the function given in Exerci… - Vidyadip

Question

Find the second order derivatives of the function given in Exercise:
$\log(\log\text{x})$

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Answer

Let $\text{y}=\log(\log\text{x})$ $\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{1}{\log\text{x}}.\frac{1}{\text{x}}=\frac{1}{\text{x}\log\text{x}}$$\therefore\ \frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{(\text{x}\log\text{x}).\frac{\text{d}}{\text{dx}}(1)-1.\frac{\text{d}}{\text{dx}}(\text{x}\log\text{x})}{(\text{x}\log\text{x})^2}$
$=\frac{(\text{x}\log\text{x}).0-1.\Big(\text{x}.\frac{1}{\text{x}}+\log\text{x}.1\Big)}{(\text{x}\log\text{x})^2}$ $=\frac{0-(1+\log\text{x})}{(\text{x}\log\text{x})}$ $=-\frac{(1+\log\text{x})}{(\text{x}\log\text{x})^2}$

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