The-derivative-of-F-x-x-2-x-3-1-log-t-dt-x-gt-0-is-

Question Number 53532 by gunawan last updated on 23/Jan/19

The derivative of F (x) = \underset{x^{2}}{\int^{x^{3}}} \frac{1}{\log t} d t

(x > 0) is

Commented by Abdo msup. last updated on 23/Jan/19

\frac{d F}{d x} (x) = \frac{{(x^{3})}^{^{'}}}{l o g (x^{3})} - \frac{{(x^{2})}^{^{'}}}{l o g (x^{2})} = \frac{3 x^{2}}{3 l o g (x)} - \frac{2 x}{2 l o g (x)}

= \frac{x^{2} - x}{l o g (x)}

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Jan/19

\frac{d F (x)}{d x} = \int_{x^{2}}^{x^{3}} \frac{\partial}{\partial x} (\frac{1}{l o g t}) d t + \frac{1}{{l o g x}^{3}} \times \frac{d}{d x} (x^{3}) - \frac{1}{{l o g x}^{2}} \times \frac{d}{d x} (x^{2})

= 0 + \frac{3 x^{2}}{3 l o g x} - \frac{2 x}{2 l o g x}

= \frac{x^{2}}{l o g x} - \frac{x}{l o g x} = \frac{x}{l o g x} (x - 1)

Commented by gunawan last updated on 23/Jan/19

thank you very much Sir

Commented by malwaan last updated on 23/Jan/19

0 how ?

Commented by tanmay.chaudhury50@gmail.com last updated on 23/Jan/19

p a r t i a l d i f f e r e n t i a t i o n o f F (t) w . r . t . x i s x i s 0

Commented by malwaan last updated on 24/Jan/19

Yes

thank you so much sir

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