Question

Diffrentiate tan^-1 1+x^2/1+x^3 w.r.t x

Answer 1

Step-by-step explanation:

Refer the attachment for the answer

Answer 2

Let tan^-1{√1+x^2 -√1-x^2 /√1+x^2 + √1-x^2}=u

Let x^2=cos2t and then

tan u = √1+cos2t - √1-cos2t / √1+cost +√1-cos2t

tan u = √1+2cos^2t-1 - √1-1+2sin^2t / √1+2cos^2t-1 + √1-1+2sin^2t

tan u = cost + sint / cost - sint

Divided by cost

tan u = 1+tant /1-tant

u= tan^-1{tan(t+π/4)}

u=(t+π/4)

t=1/2 cos^-1 x^2

u=π/4 + 1/2 cos^-1 x^2

du/dx=-x/√1-x^4

Cos^-1 x^2 =v

dv/dx=-2x/√1-x^4

du/dv=du/dx * dx/dv

du/dv=1/2

u=π/4 + 1/2 cos^-1 x^2

du/dx=1/2(-1/√1-x^