Question

If Z=tan^1 (x/y), where` x=2t, y=1-t^2, "prove that" d_z/d_t=2/(1+t^2).` 

Answer 1

`Z=tan ^-1(x/y)`     `x=2t   and    y=1-t^2` 

∴ z is the function of x and y & x and y are the functions of t. 

`Z→ tanf(x,y)→f(t)`

`Z = tan^-1 ((2t)/(1-t^2))`

Direct differentiate w.r.t t , 

`d_z/d_t=1/(1+((2t)/(1-t^2))^2xxd/dt((2t)/(1-t^2))`

=`2(1-t^2)^2/((1-t^2)^2+4t^2)xx[t.(1)/(1-t^2)^2(-2t)+1/(1-t^2)xx1]`

=` (2(1-t^2)^2)/(1+t^2)xx1/(1-t^2)^2`

∴ `d_z/d_t=2/(1+t^2)`

Hence Proved.