Real Polynomial Derivatives
Why
The real polynomials are differentiable on all
of $\R $, and have simple derivatives.
Results
Suppose $p: \R \to \R $ is a real polynomial
with coefficients $c_0, \dots , c_1, \dots , c_m$.
Then $p$ is differentiable and its derivative
$p': \R \to \R $ satisfies
\[
p'(x) = c_1 + 2c_2x + 3c_3x^2 + \cdots + mc_nx^{m-1}
\]