# 二階導數

${\displaystyle {\boldsymbol {a}}={\frac {\mathrm {d} {\boldsymbol {v}}}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}{\boldsymbol {x}}}{\mathrm {d} t^{2}}},}$

## 二階導數的冪法則

${\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\left[x^{n}\right]={\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left[x^{n}\right]={\frac {\mathrm {d} }{\mathrm {d} x}}\left[nx^{n-1}\right]=n{\frac {\mathrm {d} }{\mathrm {d} x}}\left[x^{n-1}\right]=n(n-1)x^{n-2}.}$

## 記法

${\displaystyle f''=\left(f'\right)',}$

${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}.}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)\,=\,{\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}.}$

### 其他記法

{\displaystyle {\begin{aligned}y''(x)&=\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)'\\&={\frac {(\mathrm {d} y)'\cdot \mathrm {d} x-\mathrm {d} y\cdot (\mathrm {d} x)'}{(\mathrm {d} x)^{2}}}\\&={\frac {{\frac {\mathrm {d} }{\mathrm {d} x}}(\mathrm {d} y)\cdot \mathrm {d} x-\mathrm {d} y\cdot {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {d} x}{(\mathrm {d} x)^{2}}}\\&={\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}-{\frac {\mathrm {d} y}{\mathrm {d} x}}{\frac {\mathrm {d} ^{2}x}{\mathrm {d} x^{2}}}.\end{aligned}}}

## 例

${\displaystyle f(x)=x^{3},}$

${\displaystyle f^{\prime }(x)=3x^{2}.}$

${\displaystyle f}$ 的二階導數即是對導數 ${\displaystyle f'}$ 再次求導的結果，由下式給出：

${\displaystyle f^{\prime \prime }(x)=6x.}$

${\displaystyle \sin '(x)=\cos(x),}$

${\displaystyle \sin ''(x)=\cos '(x)=-\sin(x).}$

## 與圖像的關係

### 二階導數檢驗

• ${\displaystyle f^{\prime \prime }(x)<0}$，則 ${\displaystyle f}$${\displaystyle x}$ 點取得局部極大值。
• ${\displaystyle f^{\prime \prime }(x)>0}$，則 ${\displaystyle f}$${\displaystyle x}$ 點取得局部極小值。
• ${\displaystyle f^{\prime \prime }(x)=0}$，則二階導數檢驗無定論。該點或許是拐點，也可能是極大或極小點。

## 極限

${\displaystyle f''(x)=\lim _{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}$

${\displaystyle {\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}.}$

${\displaystyle \operatorname {sgn}(x)={\begin{cases}-1,&{\text{若 }}\ x<0,\\0,&{\text{若 }}\ x=0,\\1,&{\text{若 }}\ x>0.\end{cases}}}$

{\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}}}

## 二次近似

${\displaystyle f(x)\approx f(a)+f'(a)(x-a)+{\tfrac {1}{2}}f''(a)(x-a)^{2}.}$

## 本徵值與本徵函數

${\displaystyle v_{j}(x)={\sqrt {\tfrac {2}{L}}}\sin \left({\tfrac {j\pi x}{L}}\right)}$

## 高維推廣

### 黑塞方陣

${\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}},\;{\frac {\partial ^{2}f}{\partial y^{2}}},\;{\frac {\partial ^{2}f}{\partial z^{2}}},}$

${\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}},\;{\frac {\partial ^{2}f}{\partial x\,\partial z}},\;{\frac {\partial ^{2}f}{\partial y\,\partial z}}.}$

### 拉普拉斯算子

${\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}$

## 註

1. ^ 相對之下，一階導數的記法可以較好地「當成」分數作代數運算，如鏈式法則中的抵銷。

## 參考文獻

1. ^ Content - The second derivative [目錄：二階導數]. amsi.org.au. [2020-09-16]. （原始内容存档于2022-03-24） （英语）.页面存档备份，存于互联网档案馆
2. Second Derivatives [二階導數]. Math24. [2020-09-16] （英语）.[失效連結]
3. ^ Bartlett, Jonathan; Khurshudyan, Asatur Zh. Extending the Algebraic Manipulability of Differentials [使微分更適宜代數操作]. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis. 2019, 26 (3): 217–230. （英语）.
4. ^ Editors. Reviews [評論]. Mathematics Magazine. December 20, 2019, 92 (5): 396–397. S2CID 218542586. doi:10.1080/0025570X.2019.1673628 （英语）.
5. ^ A. Zygmund. Trigonometric Series [三角級數]. Cambridge University Press. 2002: 22–23. ISBN 978-0-521-89053-3 （英语）.
6. ^ Thomson, Brian S. Symmetric Properties of Real Functions [實函數的對稱性質]. Marcel Dekker. 1994: 1. ISBN 0-8247-9230-0 （英语）.