# How do you differentiate f(x)=e^(x^2)/(e^(2x)-2x) using the quotient rule?

Nov 30, 2015

$f ' \left(x\right) = \frac{2 {e}^{{x}^{2}} \left(x {e}^{2 x} - 2 {x}^{2} - {e}^{2 x} + 1\right)}{{e}^{2 x} - 2 x} ^ 2$

#### Explanation:

According to the quotient rule:

$f ' \left(x\right) = \frac{\left({e}^{2 x} - 2 x\right) {\overbrace{\frac{d}{\mathrm{dx}} \left[{e}^{{x}^{2}}\right]}}^{2 x {e}^{{x}^{2}}} - {e}^{{x}^{2}} {\overbrace{\frac{d}{\mathrm{dx}} \left[{e}^{2 x} - 2 x\right]}}^{2 {e}^{2 x} - 2}}{{e}^{2 x} - 2 x} ^ 2$

$f ' \left(x\right) = \frac{2 x {e}^{{x}^{2}} \left({e}^{2 x} - 2 x\right) - {e}^{{x}^{2}} \left(2 {e}^{2 x} - 2\right)}{{e}^{2 x} - 2 x} ^ 2$

$f ' \left(x\right) = \frac{2 {e}^{{x}^{2}} \left(x {e}^{2 x} - 2 {x}^{2} - {e}^{2 x} + 1\right)}{{e}^{2 x} - 2 x} ^ 2$