To use the quotient rule we first identify the numerator f ( x ) = e x {\displaystyle f(x)=e^{x}} and the denominator g ( x ) = x 2 − 3 {\displaystyle g(x)=x^{2}-3} . Then we calculate f ′ ( x ) = e x {\displaystyle f'(x)=e^{x}} and g ′ ( x ) = 2 x {\displaystyle g'(x)=2x} . The rest is a walk in the park, just plug it in:
d d x ( e x x 2 − 3 ) = e x ( x 2 − 3 ) − 2 x ( e x ) ( x 2 − 3 ) 2 = e x ( x 2 − 3 − 2 x ) ( x 2 − 3 ) 2 = e x ( x − 3 ) ( x + 1 ) ( x 2 − 3 ) 2 {\displaystyle {\begin{aligned}&{\frac {d}{dx}}{\bigg (}{\frac {e^{x}}{x^{2}-3}}{\bigg )}\\=&{\frac {e^{x}(x^{2}-3)-2x(e^{x})}{(x^{2}-3)^{2}}}\\=&{\frac {e^{x}(x^{2}-3-2x)}{(x^{2}-3)^{2}}}\\=&{\frac {e^{x}(x-3)(x+1)}{(x^{2}-3)^{2}}}\end{aligned}}}