Approximation of
f
′
(
x
)
f’ (x)
by a difference quotient of the form
\[
h
−
1
[
a
1
f
(
x
+
b
1
h
)
+
a
2
f
(
x
+
b
2
h
)
+
a
3
f
(
x
+
b
3
h
)
]
{h^{ - 1}}[{a_1}f(x + {b_1}h) + {a_2}f(x + {b_2}h) + {a_3}f(x + {b_3}h)]
\]
is found to be optimized for a wide class of real-valued functions by the surprisingly asymmetric choice of
b
=
(
b
1
,
b
2
,
b
3
)
=
(
1
/
3
−
1
,
1
/
3
,
1
/
3
+
1
)
{\mathbf {b}} = ({b_1},{b_2},{b_3}) = (1/\sqrt 3 - 1,1/\sqrt 3 ,1/\sqrt 3 + 1)
. The nearly optimal choice of
b
=
(
−
2
,
3
,
6
)
{\mathbf {b}} = ( - 2,3,6)
is also discussed.