  In this paper, we consider a $L^\infty$ functional derivative estimate for
the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times
[0,T]\to\mathbb{R}$ to the Cauchy problem for scalar semi-linear parabolic
partial differential equations with a continuous nonlinearity
$f:\mathbb{R}\to\mathbb{R}$ and initial data $u_0:\mathbb{R}\to\mathbb{R}$, of
the form, \[ \sup_{x\in\mathbb{R}}|u_x (x , t)| \leq \mathcal{F}_t (f,u_0,u) \
\ \ \forall t\in [0,T] . \] Here $\mathcal{F}_t:\mathcal{A}_t\to\mathbb{R}$ is
a functional as defined in \textsection 1. We establish that the functional
derivative estimate is non-trivially sharp, by constructing a sequence
$(f_n,0,u^{(n)})$, where for each $n\in\mathbb{N}$, $u^{(n)}:\mathbb{R}\times
[0,T]\to\mathbb{R}$ is a solution to the Cauchy problem with zero initial data
and nonlinearity $f_n:\mathbb{R}\to\mathbb{R}$, and for which
$\sup_{x\in\mathbb{R}} |u_x^{(n)}(x,T)| \geq \alpha >0$, with \[
\lim_{n\to\infty} \left( \inf_{t\in [0,T]} \left(
\sup_{x\in\mathbb{R}}|u_x^{(n)}(\cdot , t)| - \mathcal{F}_t (f_n , 0 , u^{(n)})
\right) \right) = 0 . \]
