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The function is $\displaystyle{\frac{3x^4}{1-x}}$ and I am trying to find $\displaystyle{\frac{d^4}{dx^4}}$. However, I want to find the solution without using the quotient rule $4$ times in a row. I am unsure of the best alternate method. Any help is much appreciated.

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up vote 10 down vote accepted

You can use long division (or synthetic division) to see that

$$\frac{3x^4}{1 - x} = -3x^3 - 3x^2 - 3x - 3 - \frac{3}{x - 1}$$

Once you've done this, the first four terms will vanish upon differentiating $4$ times, and the last term is fairly easy to study, since it can be written as $-3 (x - 1)^{-1}$; now use the power rule and the chain rule a few times.

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