Derivative of a continuous bilinear map
= Derivative of a continuous bilinear map
{title2=$Df(a,b)(h,k)=f(h,b)+f(a,k)$}
A continuous bilinear map obeys $\|f(h,k)\|\le C\|h\|\|k\|$. Expanding $f(a+h,b+k)$ leaves the displayed linear term and remainder $f(h,k)$. For product <norm> $\rho=(\|h\|^2+\|k\|^2)^{1/2}$, the remainder has <norm> at most $C\rho^2/2$, hence is $o(\rho)$. This proves the <Fréchet derivative> formula.