Answer
$$\frac{{dy}}{{dx}} = 2\tanh x{\operatorname{sech} ^2}x$$
Work Step by Step
$$\eqalign{
& y = {\tanh ^2}x \cr
& y = {\left( {\tanh x} \right)^2} \cr
& {\text{computing }}dy/dx \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}{\left( {\tanh x} \right)^2} \cr
& {\text{by the chain rule}} \cr
& \frac{{dy}}{{dx}} = 2{\left( {\tanh x} \right)^{2 - 1}}\frac{d}{{dx}}\left( {\tanh x} \right) \cr
& \frac{{dy}}{{dx}} = 2{\left( {\tanh x} \right)^{2 - 1}}\left( {{{\operatorname{sech} }^2}x} \right) \cr
& {\text{multiplying}} \cr
& \frac{{dy}}{{dx}} = 2\tanh x{\operatorname{sech} ^2}x \cr} $$