Question

Find the derivatives of the following:

If u = `tan^-1  (sqrt(1 + x^2) - 1)/x` and v = `tan^-1 x`, find `("d"u)/("d"v)`

Answer 1

u = `tan^-1  (sqrt(1 + x^2) - 1)/x` 

Put x = tan θ,

u = `tan^-1  ((sqrt(1 + tan^2theta) - 1)/tan theta)`

= `tan^-1 ((sqrt(sec^2theta) - 1)/tan theta)`

= `tan^-1 ((sectheta - 1)/tantheta)`

= `tan^-1 ((1/(costheta) - 1)/((sintheta)/costheta))`

= `tan^-1  (((1 - costheta)/costheta)/((sin theta)/(costheta)))`

= `tan^-1 ((1 - cos theta)/sintheta)`

= `tan^-1 ((2 sin^2  theta/2)/(2 sin  theta/2  cos  theta/2))`

= `tan^-1 ((sin  theta/2)/(cos  theta/2))`

= `tan^-1 (tan  theta/2)`

= `theta/2`

u = `1/2 tan^-1 (x)`

`("d"u)/("d"x) = 1/2 xx 1/(1 + x^2)`  .......(1)

Let v = `tan^-1 (x)`

`("d"v)/("d"x) = 1/(1 + x^2)`  ........(2)

From equations (1) and (2)

`(("d"u)/("d"x))/(("d"v)/("d"x)) = (1/(2(1 + x^2)))/(1/(1 + x^2))`

`("d"u)/("d"v) = 1/2`

`("d"(tan^-1  ((sqrt(1 + x^2) - 1)/x)))/("d"(tan^-1 x)) = 1/2`