Answer
$$\frac{{dy}}{{dx}} = x{\operatorname{sech} ^2}x + \tanh x$$
Work Step by Step
$$\eqalign{
& y = x\tanh x \cr
& {\text{computing }}dy/dx \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {x\tanh x} \right) \cr
& {\text{by the product rule}} \cr
& \frac{{dy}}{{dx}} = x\frac{d}{{dx}}\left( {\tanh x} \right) + \tanh x\frac{d}{{dx}}\left( x \right) \cr
& {\text{using basic formulas for differentiation}} \cr
& \frac{{dy}}{{dx}} = x\left( {{{\operatorname{sech} }^2}x} \right) + \tanh x\left( 1 \right) \cr
& {\text{multiplying}} \cr
& \frac{{dy}}{{dx}} = x{\operatorname{sech} ^2}x + \tanh x \cr} $$
