连续两次用一阶导数的幂法则
d
2
d
x
2
[
x
n
]
=
d
d
x
d
d
x
[
x
n
]
=
d
d
x
[
n
x
n
−
1
]
=
n
d
d
x
[
x
n
−
1
]
=
n
(
n
−
1
)
x
n
−
2
.
{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\left[x^{n}\right]={\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left[x^{n}\right]={\frac {\mathrm {d} }{\mathrm {d} x}}\left[nx^{n-1}\right]=n{\frac {\mathrm {d} }{\mathrm {d} x}}\left[x^{n-1}\right]=n(n-1)x^{n-2}.}
公式对任意实数
n
{\displaystyle n}
f
(
x
)
=
sin
(
2
x
)
{\displaystyle f(x)=\sin(2x)}
x
{\displaystyle x}
−
π
/
4
{\displaystyle -\pi /4}
5
π
/
4
{\displaystyle 5\pi /4}
0
,
π
/
2
,
π
{\displaystyle 0,\ \pi /2,\ \pi }
函数
f
{\displaystyle f}
凹性 （concavity ）。[ 2] 向上凹 （向上弯，又称为凸函数或下凸函数），意即其切线 总位于图像下方“承托”。反之，若二阶导数在某区间恒负，则函数在该区间向下凹 （向下弯，又称为凹函数或上凸函数），其切线总位于图像的上方“压制”着。
二阶导数若存在，则可以只用一个极限 写出：
f
″
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
2
f
(
x
)
+
f
(
x
−
h
)
h
2
.
{\displaystyle f''(x)=\lim _{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}
以上极限称为二阶对称导数 [ 5] [ 6]
右侧欲求极限的分式，可理解成差商
f
(
x
+
h
)
−
2
f
(
x
)
+
f
(
x
−
h
)
h
2
=
f
(
x
+
h
)
−
f
(
x
)
h
−
f
(
x
)
−
f
(
x
−
h
)
h
h
.
{\displaystyle {\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}.}
故其极限可视作序列 二阶差分 的连续版本。
然而，上述极限存在并不推出函数
f
{\displaystyle f}
符号函数
sgn
{\displaystyle \operatorname {sgn} }
sgn
(
x
)
=
{
−
1
,
若
x
<
0
,
0
,
若
x
=
0
,
1
,
若
x
>
0.
{\displaystyle \operatorname {sgn}(x)={\begin{cases}-1,&{\text{若 }}\ x<0,\\0,&{\text{若 }}\ x=0,\\1,&{\text{若 }}\ x>0.\end{cases}}}
符号函数在原点不连续，从而不可导，尤其并非二阶可导。但是，在
x
=
0
{\displaystyle x=0}
lim
h
→
0
sgn
(
0
+
h
)
−
2
sgn
(
0
)
+
sgn
(
0
−
h
)
h
2
=
lim
h
→
0
sgn
(
h
)
−
2
⋅
0
+
sgn
(
−
h
)
h
2
=
lim
h
→
0
sgn
(
h
)
+
(
−
sgn
(
h
)
)
h
2
=
lim
h
→
0
0
h
2
=
0.
{\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)-2\cdot 0+\operatorname {sgn}(-h)}{h^{2}}}\\&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)+(-\operatorname {sgn}(h))}{h^{2}}}=\lim _{h\to 0}{\frac {0}{h^{2}}}=0.\end{aligned}}}
二阶导数的高维推广，其一是同时考虑全体二阶偏导数
∂
2
f
∂
x
i
∂
x
j
{\displaystyle {\tfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}}
f
:
R
3
→
R
{\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }
∂
2
f
∂
x
2
,
∂
2
f
∂
y
2
,
∂
2
f
∂
z
2
,
{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}},\;{\frac {\partial ^{2}f}{\partial y^{2}}},\;{\frac {\partial ^{2}f}{\partial z^{2}}},}
以及混合偏导数
∂
2
f
∂
x
∂
y
,
∂
2
f
∂
x
∂
z
,
∂
2
f
∂
y
∂
z
.
{\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}},\;{\frac {\partial ^{2}f}{\partial x\,\partial z}},\;{\frac {\partial ^{2}f}{\partial y\,\partial z}}.}
还有其他次序的混合偏导数，如
∂
2
f
∂
y
∂
x
{\displaystyle {\tfrac {\partial ^{2}f}{\partial y\,\partial x}}}
二阶导数的对称性 ，只要
f
{\displaystyle f}
对称方阵 ，称为黑塞方阵 （英语：Hessian 或Hessian matrix ）。该方阵的本征值 适用于多变量情况的二阶导数检验（称为二阶偏导数检验
另一种常见推广，则是只考虑对同一个变量的二阶导数，再求和，得到拉普拉斯算子 （Laplace operator 或Laplacian ）。拉氏微分算子记作
∇
2
{\displaystyle \nabla ^{2}}
Δ
{\displaystyle \Delta }
∇
2
f
=
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
.
{\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}
函数的拉氏算子等于梯度 的散度 ，亦是前述黑塞方阵之迹 。
Bartlett, Jonathan; Khurshudyan, Asatur Zh. Extending the Algebraic Manipulability of Differentials [使微分更适宜代数操作]. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis. 2019, 26 (3): 217–230. arXiv:1801.09553 （英语） . A. Zygmund. Trigonometric Series [三角级数]. Cambridge University Press. 2002: 22–23. ISBN 978-0-521-89053-3（英语） .
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