偏導數

偏導數者，乃多元函數${\displaystyle f(x_{1},x_{2},...,x_{i},...,x_{n})}$於某變量${\displaystyle x_{i}}$之變率。

計偏導數時，若微分於某變量${\displaystyle x_{i}}$（${\displaystyle {\frac {\partial f}{\partial x_{i}}}}$），則視其余變量${\displaystyle x_{j}}$（${\displaystyle j\in \mathbb {Z} \cup [1,n]\not i}$）者為常數，僅以${\displaystyle x_{i}}$作變量運之。

定義

設${\displaystyle U}$為一開區間且函數${\displaystyle f:U\to R}$，${\displaystyle f}$之偏微分於變量${\displaystyle x_{i}}$，於點${\displaystyle a:(a_{1},a_{2},...a_{i},...,a_{n})\in U}$之偏導數定義為： ${\displaystyle {\frac {\partial }{\partial x_{i}}}f(a_{1},a_{2},...,a_{i},...,a_{n})=\lim _{\Delta x_{i}\to 0}{\frac {f(a_{1},a_{2},...,a_{i}+\Delta x_{i},...,a_{n})-f(a_{1},a_{2},...,a_{i},...,a_{n})}{\Delta x_{i}}}}$

例

使 ${\displaystyle z=x^{2}+xy+y^{2}}$ 。

視 ${\displaystyle y}$ 作常數， ${\displaystyle {\frac {\partial z}{\partial x}}=2x+y}$ 。

視 ${\displaystyle x}$ 作常數， ${\displaystyle {\frac {\partial z}{\partial y}}=2y+x}$ 。

據

https://www.mathsisfun.com/calculus/derivatives-partial.html