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The derivative of any polynomial function of one variable is easily obtained. For the trivial case of f(x) = a, for some constant a (a degree 0 polynomial):

$\frac{d}{dx}a=0$ [Proof]

For any real-valued differentiable functions f(x) and g(x):

$\frac{d}{dx}a\,f(x)=a\,f'(x)$ [Proof]

$\frac{d}{dx}(f(x)+g(x))=f'(x)+g'(x)$ [Proof]

And for any real number r:

$\frac{d}{dx}x^r=rx^{r-1}$ [Proof]

Which covers any single variable polynomial function. Derivatives of non-polynomial functions require additional rules.

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