If `x=exp{tan^(-1)((y-x^(2))/(x^(2)))}`, then `(dy)/(dx)` equal

Options

A. `2x[1+tan(logx)]+sec^(2)(logx)`
B. `x[1+tan(logx)]+sec^(2)(logx)`
C. `2x[1+tan(logx)]+x^(2)sec^(2)(logx)`
D. `2x[1+tan(logx)]+sec^(2)(logx)`

Answer

A

Explanation

Gilven that
`x="exp" [tan^(-1)((y-x^(2))/(x^(2)))]`
Taking log on both sides we get
`logx=tan^(-1)((y-x^(2))/(x^(2)))`
`implies(y-x^(2))/(x^(2))=tan(logx)`
`impliesy=x^(2)tan(logx)+x^(2)`
On differenting w.r.t. x, we get
`(dy)/(dx)=2xtan(logx)`
`+x^(2)(sec^(2)(logx))/x+2x`
`implies(dy)/(dx)=2x tan (logx)`
`+x sec^(2)(logx)+2x`
`implies(dy)/(dx)=2x[1+tan(logx)]`
`+sec^(2)(logx)`