Chain rule time here!
But first let's just remember that tan^-1(x) = arctan(x).
Rewriting your expression, we get y = arctan(7x^2-1)^(1/2)
We can name v=7x^2-1 and u=v^(1/2). Thus, y=arctan(u).
So, now, by chain rule definition,
(dy)/(du)(du)/(dv)(dv)/(dx) = (dy)/(dx)
Going part by part, here:
(dy)/(du) = (u')/(1+u^2)
(du)/(dv)=(1/2)*(1/v^(1/2))
(dv)/(dx)=14x
Now, getting back to the chain rule definition, we have this:
(u')/(1+u^2)(1/cancel(2))(1/v^(1/2))cancel(14)7x
After cancelling the 2 and simplifying 14 as 7, we can now substitute the values of u and v, as our objective is to get the answer as function of x.
((1/2)*(1/v^(1/2)))/(1+(v^(1/2))^2)*1/(7x^2-1)*7x
(1/2(7x^2-1)^(1/2))/(cancel(1)+7x^2cancel(-1))*1/(7x^2-1)*7x
We can proceed to some more cancellings:
(1/2(7x^2-1)^(1/2))/(7x^2)*1/(7x^cancel(2)-1)*cancel(7x)
And, now, rearrange:
((7x^2-1)^(1/2)/2)/(7x^2(7x-1))=color(green)((7x^2-1)^(1/2))/color(green)(14x^2(7x-1)
