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"If for "x in (0,(1)/(4))," the derivati...

`"If for "x in (0,(1)/(4))," the derivative of "tan^(-1)((6xsqrt(x))/(1-9x^(3)))" is "sqrt(x)cdotg(x),` then g(x) equals

A

`9/(1+9x^(3))`

B

`(3xsqrtx)/(1-9x^(3))`

C

`(3x)/(1-9x^(3))`

D

`3/(1+9x^(3))`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `y=tan^(-1)((6xsqrtx)/(1-9x^(3)))= tan^(-1)[(2*(3x^(3//2)))/(1-(3x^(3//2))^(2))]`
`=2tan^(-1) (3x^(3//2))[:' 2 tan^(-1) x = tan^(-1)(2x)/(1-x^(2))]`
`:. (dy)/(dx) =2 * 1/(1+(3x^(3//2))^(2))* 3 xx 3/2 (x)^(1//2) = 9/(1+9x^(3))*sqrtx`
` :." " g(x) = 9/(1+9x^(3)) `
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