Maybe it is a soft question, I can't find any information about this. How people understood that we can use derivative shortcuts instead of calculating limits and how did they derive them?
By shortcuts I mean rules like $(x^n)^{'} = nx^{n-1}$. I find them somehow intuitive, like if we have a linear function, its derivative should be constant, etc. But I can't figure out how one can derive those shortcuts. Any references, links will be appreciated.
P.S. I am also interested in historical part who found them. Were they proved at once by someone? Did Newton or Leibniz know these shortcuts?
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$\begingroup$You can compute directly from the definition of the derivative. For your power rule example, there are some ways to do this, none of which are L'Hopital's rule.
$$ \lim_{h\to 0}\frac{1}{h}((x+h)^n-x^n)\\ \stackrel{\text{binomial formula}}{=} \lim_{h\to 0}\frac{1}{h}\left(x^n-x^n+nx^{n-1}h+h^2\sum_{k=0}^n{n \choose k}x^{n-k}h^{k-2}\right)\\ =nx^{n-1}+\lim_{h\to 0}h\sum_{k=0}^n{n \choose k}x^{n-k}h^{k-2}=nx^{n-1} $$ the derivative of $f(x)=x^n$ at $x$.
$\endgroup$ 1 $\begingroup$If you can take limits on tangents to the line $x^2$ as you approach some point on the line then you are virtually there.
Rather than just calculate limits for individual functions such as $x^2$ to get $\dfrac{dx^2}{dx}=2x$ all you need do is repeat the same for general forms such as $x^n$ which yields the general form of the derivative $\dfrac{dx^n}{dx}=nx^{n-1}$.
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