二元函数方向导数的计算（B013）

问题

若已知函数 $f(x, y)$ 在点 $\left(x_{0}, y_{0}\right)$ 处可微, 且 $(\cos \alpha, \cos \beta)$ 是 $\boldsymbol{l}$ 方向的方向余弦.
那么，该函数在该点沿任何方向 $\boldsymbol{l}$ 的方向导数 $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ 都存在，则 $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $?$

选项

[A].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \beta$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \cos \alpha$

[B].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \sin \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \sin \beta$

[C].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \cos \beta$

[D].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \alpha$ $-$ $f_{y}\left(x_{0}, y_{0}\right) \cos \beta$

   答 案   

$\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \cos \beta$