\[\frac{d}{dx} \frac{1}{{e}^{x}{x}^{3}}\]
1
Expand fraction.
ddx1ex×1x3\frac{d}{dx} \frac{1}{{e}^{x}}\times \frac{1}{{x}^{3}}

2
Use Product Rule to find the derivative of 1ex×1x3\frac{1}{{e}^{x}}\times \frac{1}{{x}^{3}}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
ddx1exx3+ddx1x3ex\frac{\frac{d}{dx} \frac{1}{{e}^{x}}}{{x}^{3}}+\frac{\frac{d}{dx} \frac{1}{{x}^{3}}}{{e}^{x}}

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−1exx3+ddx1x3ex-\frac{1}{{e}^{x}{x}^{3}}+\frac{\frac{d}{dx} \frac{1}{{x}^{3}}}{{e}^{x}}

4
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−1exx3−3x4ex-\frac{1}{{e}^{x}{x}^{3}}-\frac{3}{{x}^{4}{e}^{x}}

Done
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