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Maybe I'm just having a brain breakdown moment, but it seems weird to me that both functions have exactly the same derivative, namely $\frac{1}{(1-x)^2}$.

Obviously I'm not disputing whether or not it's correct, but I'm looking for some sort of intuition/justification (geometric or otherwise) as to why this is indeed the case.

I can take the derivatives manually so this is really not a question of not understanding the algebra, rather maybe having poor geometric intuition as to why the changing numerator as a function of $x$ doesn't impact the rate of change of the overall function.

2 Answers 2

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Note that $\cfrac 1{1-x}-1=\cfrac x{1-x}$ so the two differ by a constant and will have the same derivative.

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    Shame on me for not seeing this immediately. Somehow it just seemed wrong to have a value changing as a function of $x$ in the numerator without any impact on the rate of change. – 2017-01-25
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Hint:

$$\frac x{1-x}=-1+\frac1{1-x}$$

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