Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (d)/Solution 1

To use the quotient rule we first identify the numerator $f(x)=e^{x}$ and the denominator $g(x)=x^{2}-3$. Then we calculate $f'(x)=e^{x}$ and $g'(x)=2x$. The rest is a walk in the park, just plug it in:

${\begin{aligned}&{\frac {d}{dx}}{\bigg (}{\frac {e^{x}}{x^{2}-3}}{\bigg )}\\=&{\frac {e^{x}(x^{2}-3)-2x(e^{x})}{(x^{2}-3)^{2}}}\\=&{\frac {e^{x}(x^{2}-3-2x)}{(x^{2}-3)^{2}}}\\=&{\frac {e^{x}(x-3)(x+1)}{(x^{2}-3)^{2}}}\end{aligned}}$