$f(x)=-x^3+6x^2+66x+666$
equation of slope at point $x$, $m=f'(x)=-3x^2+12x+66$
to find max/min value of slope, put $\frac{dm}{dx}=0$
$\frac{dm}{dx}=-6x+12=0 , x=2$
now, $\frac{d^2m}{dx^2}=-6<0$, therefore at $x=2$, slope m is maximum.
therefore max slope at $x=2$ is $m_{max}=78$