Linearity of the derivative operator of polynomials

Question

Suppose $f(x)=x^i$ as a simple polynomial function of power $i$. As we know from the definition of the derivative operators, such an operator must be linear. How can we prove that the derivative operator of polynomials is actually a linear operator?
I am specifically interested in using Fréchet derivative formula to give a proof.

Answer 1

\begin{align} &\lim_{h \to 0} \frac{|r(x+h)^i+s(x+h)^j-(rx^i+sx^j)-(irx^{i-1}-jsx^{j-1})h|}{|h|}\\ &=\lim_{h \to 0}\left|r\left(\frac{(x+h)^i-x^i}{h} -ix^{i-1}\right)+s\left(\frac{(x+h)^j-x^j}{h} -jx^{j-1}\right)\right|\\ &=\left|r\lim_{h \to 0}\left(\frac{(x+h)^i-x^i}{h} -ix^{i-1}\right)+s\lim_{h \to 0}\left(\frac{(x+h)^j-x^j}{h} -jx^{j-1}\right)\right|\\ &=0 \end{align}

where I have used continuity of norm and linearity of limit.