Differentiating the Reciprocal of a Function

1. \(\dfrac{d}{dx}\!\left( \dfrac{1}{x^{2}+x}\right) \underset{\underset{\color{#0066A7}{\text{Use (1).}}}{\color{#0066A7}{\uparrow}}}{=} -\dfrac{\dfrac{d}{dx}( x^{2}+x) }{( x^{2}+x) ^{2}}=-\dfrac{2x+1}{( x^{2}+x) ^{2}}\)
2. \(\dfrac{d}{dx}e^{-x}=\dfrac{d}{dx}\!\left( \dfrac{1}{e^{x}}\right) \underset{\underset{\color{#0066A7}{\text{Use (1).}}}{\color{#0066A7}{\uparrow}}}{=} -\dfrac{\dfrac{d }{dx}e^{x}}{( e^{x}) ^{2}}=-\dfrac{e^{x}}{e^{2x}}=-\dfrac{1}{e^{x} }=-e^{-x}\)