Question

What is the derivative of? : `y=3tan^(-1)(x+sqrt(1+x^2) )`

Answer 1

`dy/dx = ( 3(1+x/(sqrt(1+x^2))))/(1+(x+sqrt(1+x^2))^2)`

Explanation:

We seek (I assume):

`dy/dx` where `y=3tan^(-1)(x+sqrt(1+x^2) )`

We will need the standard result:

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

And the power rule, in conjunction with the chain rule, for:

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

Then we can apply the product rule to the original function to get:

`dy/dx = 3 {1/(1+(x+sqrt(1+x^2))^2) } d/dx (x+sqrt(1+x^2) )`
`\ \ \ \ \ = 3 {1/(1+(x+sqrt(1+x^2))^2) } {1+x/(sqrt(1+x^2))}`
`\ \ \ \ \ = ( 3(1+x/(sqrt(1+x^2))))/(1+(x+sqrt(1+x^2))^2)`