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Question
The derivative of \[f\left( x \right) = \int\limits_{x^2}^{x^3} \frac{1}{\log_e t} dt, \left( x > 0 \right),\] is
Options
- \[\frac{1}{3 \ln x}\]
- \[\frac{1}{3 \ln x} - \frac{1}{2 \ln x}\]
(ln x)−1 x (x − 1)
- \[\frac{3 x^2}{\ln x}\]
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Solution
(ln x)−1 x (x − 1)
Using Newton Leibnitz formula
\[f' (x) = \frac{1}{\log_e x^3}(3 x^2 ) - \frac{1}{\log_e x^2}(2x) \]
\[= \frac{3 x^2}{3\ln x}- \frac{2x}{2\ln x} \]
\[= \frac{x^2}{\ln x} - \frac{x}{\ln x} \]
\[= \frac{1}{\ln x}x(x - 1) \]
\[= {(\ln x)}^{- 1} x(x - 1)\]
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