偏導數者,乃多元函數f(x1,x2,...,xi,...,xn){\displaystyle f(x_{1},x_{2},...,x_{i},...,x_{n})}於某變量xi{\displaystyle x_{i}}之變率。
計偏導數時,若微分於某變量xi{\displaystyle x_{i}}(∂f∂xi{\displaystyle {\frac {\partial f}{\partial x_{i}}}}),則視其余變量xj{\displaystyle x_{j}}(j∈Z∪[1,n]i̸{\displaystyle j\in \mathbb {Z} \cup [1,n]\not i})者為常數,僅以xi{\displaystyle x_{i}}作變量運之。
設U{\displaystyle U} 為一開區間且函數f:U→R{\displaystyle f:U\to R} ,f{\displaystyle f} 之偏微分於變量xi{\displaystyle x_{i}} ,於點a:(a1,a2,...ai,...,an)∈U{\displaystyle a:(a_{1},a_{2},...a_{i},...,a_{n})\in U} 之偏導數定義為: ∂∂xif(a1,a2,...,ai,...,an)=limΔxi→0f(a1,a2,...,ai+Δxi,...,an)−f(a1,a2,...,ai,...,an)Δxi{\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}}}}
使 z=x2+xy+y2{\displaystyle z=x^{2}+xy+y^{2}} 。
視 y{\displaystyle y} 作常數, ∂z∂x=2x+y{\displaystyle {\frac {\partial z}{\partial x}}=2x+y} 。
視 x{\displaystyle x} 作常數, ∂z∂y=2y+x{\displaystyle {\frac {\partial z}{\partial y}}=2y+x} 。
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