Often in Physics or Engineering it is necessary to use a calculus operation known as differentiation . Unlike textbook mathematics, the differentiated functions are data generated by an experiment or a computer code.
Taylor series as seen in Equation 1.
f
(
x
+
h
)
=
f
(
x
)
+
f
′
(
x
)
h
+
f
(
2
)
(
x
)
2
!
h
2
+
f
(
3
)
(
x
)
3
!
h
3
+
⋯
(
1
)
{\displaystyle f(x+h)=f(x)+f^{'}(x)h+{\frac {f^{(2)}(x)}{2!}}h^{2}+{\frac {f^{(3)}(x)}{3!}}h^{3}+\cdots \quad (1)}
Next by cutting off the Taylor series after the fourth term and evaluating it at h and -h yields Equations (2) and (3).
f
(
x
+
h
)
=
f
(
x
)
+
f
′
(
x
)
h
+
f
(
2
)
(
x
)
2
!
h
2
+
f
(
3
)
(
c
1
)
3
!
h
3
(
2
)
{\displaystyle f(x+h)=f(x)+f^{'}(x)h+{\frac {f^{(2)}(x)}{2!}}h^{2}+{\frac {f^{(3)}(c_{1})}{3!}}h^{3}\quad (2)}
f
(
x
−
h
)
=
f
(
x
)
−
f
′
(
x
)
h
+
f
(
2
)
(
x
)
2
!
h
2
−
f
(
3
)
(
c
2
)
3
!
h
3
(
3
)
{\displaystyle f(x-h)=f(x)-f^{'}(x)h+{\frac {f^{(2)}(x)}{2!}}h^{2}-{\frac {f^{(3)}(c_{2})}{3!}}h^{3}\quad (3)}
Then by subtracting Equation (2) by Equation (3) yields.
f
(
x
+
h
)
−
f
(
x
−
h
)
=
2
f
′
(
x
)
h
+
f
(
3
)
(
c
1
)
3
!
h
3
+
f
(
3
)
(
c
2
)
3
!
h
3
{\displaystyle f(x+h)-f(x-h)=2f^{'}(x)h+{\frac {f^{(3)}(c_{1})}{3!}}h^{3}+{\frac {f^{(3)}(c_{2})}{3!}}h^{3}}
f
′
(
x
)
=
f
(
x
+
h
)
−
f
(
x
−
h
)
2
h
+
O
(
h
2
)
{\displaystyle f^{'}(x)={\frac {f(x+h)-f(x-h)}{2h}}+O(h^{2})}
f
′
(
x
)
=
f
(
x
+
h
)
−
f
(
x
)
h
+
O
(
h
)
{\displaystyle f^{'}(x)={\frac {f(x+h)-f(x)}{h}}+O(h)}
f
′
(
x
)
=
f
(
x
)
−
f
(
x
−
h
)
h
+
O
(
h
)
{\displaystyle f^{'}(x)={\frac {f(x)-f(x-h)}{h}}+O(h)}
The second order derivatives can be obtained by adding equations (2) and (3) (if properly expanded to include the fourth-derivative-term):
f
″
(
x
)
=
f
(
x
+
h
)
−
2
f
(
x
)
+
f
(
x
−
h
)
h
2
+
O
(
h
2
)
{\displaystyle f^{''}(x)={\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}+O(h^{2})}
