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I am reading Calculus Made Easy where in Chapter IV: $$(x+dx)^{-2}$$ Is refactored as: $$x^{-2}\left(1+\frac{dx}x\right)^{-2}$$ Could someone give me an insight into this refactoring? I can see from this question Negative Exponents in Binomial Theorem the equation $(a + b)^n = a^n(1 + \frac{b}{a})^n $ though I don't know if this rule has a name so that I may research it further.

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$$(a+b)^n=\left(a\left(1+\frac{b}{a}\right)\right)^n=(a)^n\left(1+\frac{b}{a}\right)^n=a^n\left(1+\frac{b}{a}\right)^n$$ In your question you have: $a=x$, $b=dx$ and $n=-2$

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No need for any sort of binomial theorem. Just factor out an $x$, as follows

$$(x+dx)^{-2} = (x(1+\frac{dx}{x}))^{-2} = x^{-2} (1+\frac{dx}{x})^{-2}.$$

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$\textbf{Hint:}$Write $(x+dx)^{-2}$ as $\displaystyle \frac{1}{(x+dx)^{2}}$ and use your knowledge.

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