`"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

Options

A. `(9)/(1+9x^(3))`
B. `(3xsqrt(x))/(1-9x^(3))`
C. `(3x)/(1-9x^(3))`
D. `(3)/(1+9x^(3))`

Answer

A

Explanation

Let `y=tan^(-1)((6xsqrt(x))/(1-9x^(3))),x in(0,(1)/(4))`. Then,
`y=tan^(-1)((2xx3xsqrt(x))/(1-(3xsqrt(x))^(2)))`
`implies" "y=2tan^(-1)(3xsqrt(x))" "[{:(becausetan^(-1)((2x)/(1-x^(2)))=2tan^(-1)x","),(" if "-1ltxlt1):}]`
`implies" "(dy)/(dx)=2xx(1)/(1+9x^(3))xx(9)/(2)sqrt(x)=sqrt(x)(9)/(1+9x^(3))`
`implies" "(dy)/(dx)=sqrt(x)g(x)," where "g(x)=(1)/(1+9x^(3))`