Prove there is no polynomial with derivative 1/x

by RoRi

October 14, 2015

Prove that there is no polynomial $f$ such that

$f'(x) = \frac{1}{x}.$

Proof. We know from Example 1 of Section 4.5 in Apostol (p. 166) that every polynomial is differentiable everywhere on $\mathbb{R}$ . (In that example we show that the derivative of a polynomial is a polynomial, and we know that polynomials are defined everywhere on $\mathbb{R}$ .) However, the function $\frac{1}{x}$ is not defined for $x = 0$ . Hence, this function cannot be the derivative of a polynomial $. \qquad \blacksquare$

Stumbling Robot

Prove there is no polynomial with derivative 1/x

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