用第一性原理计算导数
差商不仅可以直接用于计算一阶导数,它也可以直接用于高价导数的计算. 先考虑函数 g 及其相关的差商.
In[1]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_28.png)
g[x_] := x^2 Exp[x]
In[2]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_29.png)
dq1[x_] = DifferenceQuotient[g[x], {x, h}]
Out[2]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_26.png)
对差商取极限即给出一阶导数.
In[3]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_30.png)
Limit[dq1[x], h -> 0]
Out[3]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_27.png)
In[4]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_31.png)
Limit[dq1[x], h -> 0];
Simplify[% == g'[x]]
Out[4]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_28.png)
可以直接根据二阶差商来计算二阶导数,而不必去参引一阶导数.
In[5]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_32.png)
dq2[x_] = DifferenceQuotient[g[x], {x, 2, h}]
Out[5]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_29.png)
时的极限便是二阶导数.
In[6]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_33.png)
Limit[dq2[x], h -> 0]
Out[6]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_30.png)
In[7]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_34.png)
Limit[dq2[x], h -> 0];
Simplify[% == g''[x]]
Out[7]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_31.png)
g 的一阶导数的差商与二阶差商是不同的函数,但它的极限也是二阶导数.
In[8]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_35.png)
dqp[x_] = DifferenceQuotient[g'[x], {x, h}]
Out[8]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_32.png)
In[9]:=
![Click for copyable input](assets.zh/evaluate-a-derivative-using-first-principles/In_36.png)
Limit[dqp[x], h -> 0]
Out[9]=
![](assets.zh/evaluate-a-derivative-using-first-principles/O_33.png)