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Derivative formulas

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The derivative of any polynomial function of one variable is easily obtained. If c \in \mathbb R (or a constant function) and f,g:D \rightarrow \mathbb R are both differentiable on some set D', then so are cf, f+g, -f, and fg. If, in addition, g is nonzero on D', then \frac 1 g (and also \frac f g) are differentiable on D'. Also, if f is differentiable on g\left(D'\right), then f \circ g is differentiable on D'. For the trivial case of f(x) = a, for some constant a (a degree 0 polynomial):

a'=0 [Proof]

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

And for any real number r:

(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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