  Suppose that $\mathscr{A}$ is an operator algebra on a Hilbert space $H$. An
element $V$ in $\mathscr{A}$ is called an all-derivable point of $\mathscr{A}$
for the strong operator topology if every strong operator topology continuous
derivable mapping $\phi$ at $V$ is a derivation. Let $\mathscr{N}$ be a
complete nest on a complex and separable Hilbert space $H$. Suppose that $M$
belongs to $\mathscr{N}$ with $\{0\}\neq M\neq\ H$ and write $\hat{M}$ for $M$
or $M^{\bot}$. Our main result is: for any $\Omega\in alg\mathscr{N}$ with
$\Omega=P(\hat{M})\Omega P(\hat{M})$, if $\Omega |_{\hat{M}}$ is invertible in
$alg\mathscr{N}_{\hat{M}}$, then $\Omega$ is an all-derivable point in
$alg\mathscr{N}$ for the strong operator topology.
