  Given a set $\Sigma$ of equations, the free-algebra functor $F_{\Sigma}$
associates to each set $X$ of variables the free algebra $F_{\Sigma}(X)$ over
$X$. Extending the notion of \emph{derivative} $\Sigma'$ for an arbitrary set
$\Sigma$ of equations, originally defined by Dent, Kearnes, and Szendrei, we
show that $F_\Sigma$ preserves preimages if and only if $\Sigma \vdash
\Sigma'$, i.e. $\Sigma$ derives its derivative $\Sigma'$. If $F_\Sigma$ weakly
preserves kernel pairs, then every equation $p(x,x,y)=q(x,y,y)$ gives rise to a
term $s(x,y,z,u)$ such that $p(x,y,z)=s(x,y,z,z)$ and $q(x,y,z)=s(x,x,y,z)$. In
this case n-permutable varieties must already be permutable, i.e. Mal'cev.
Conversely, if $\Sigma$ defines a Mal'cev variety, then $F_\Sigma$ weakly
preserves kernel pairs. As a tool, we prove that arbitrary $Set-$endofunctors
$F$ weakly preserve kernel pairs if and only if they weakly preserve pullbacks
of epis.
