Example 1. Derive the equations of the tangent and normal lines to the curve $y=x \ln x$, drawn at the point with abscissa $x=e$.
Solution. We have: $x_{0}=e, y_{0}=y(e)=e, y^{\prime}=\ln x+1, k=y^{\prime}(e)=2$,

$(t): y=e+2(x-e),(n): y=e-\frac{1}{2}(x-e)(\text{cm} . \S 2)$.

Answer. $(t): y=2 x-e,(n): y=-\frac{1}{2} x+\frac{3}{2} e$.