Find the derivative of f(x)=exp((tanx)^(1/2))

We use the chain rule. Let u(x)=exp(x), v(x)=x^1/2, w(x) = tan(x).
Then f(x) = u(v(w(x))). So by the chain rule, f'(x) = u'(v(w(x)))*(v(w(x)))'.
u'(x) = exp(x).
By the chain rule, (v(w(x)))' = v'(w(x))w'(x).
v'(x) = x^-1/2/2w'(x)= sec²(x)
So f'(x) = exp((tan(x)^1/2))(cos(x)/2)