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Question
Find the second order derivative of the function.
log (log x)
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Solution
Let, y = log (log x)
Differentiating both sides with respect to x,
`dy/dx = d/dx log (log x)`
= `1/(log x). d/dx (log x)`
= `1/(log x) xx 1/x`
= `1/(x log x)`
= (x log x)−1
Differentiating both sides again with respect to x,
`d/dx [dy/dx] = d/dx [(x log x)^-1]`
`(d^2y)/dx^2 = -1 (x log x)^(-1-1) d/dx (x log x)`
= `-1 (x log x)^-2 [x d/dx (log x) + log x d/dx (x)]`
= `-1 xx 1/(x log x)^2 [x xx 1/x + log x (1)]`
= `(-(1 + log x))/ (x log x)^2`
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