Sharp central-secant derivative error ($|f^{\prime }(x)-(f(x+h)-f(x-h))/(2h)|\le h^2\Vert f^{\prime \prime \prime }{\Vert }_{\infty }/6$)

For $f\in C^3[x-h,x+h]$, the Peano kernel for the error of the central secant slope has one sign. Integrating its absolute value gives the displayed constant $h^2/6$, and a cubic polynomial attains it. At $x=1$, $h=1$, the unnormalized order-two kernel is $-{\theta }^2/2$ on $[0,1]$ and $-(2-\theta {)}^2/2$ on $[1,2]$; the integral representation includes the factor $1/2!$.