방향 도함수 $D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}입니다.
기울기 벡터 \nabla f:
\displaystyle \nabla f = \langle 2xy,\; x^2\rangle,\ \nabla f(1,2)=\langle 4,\,1\rangle.
단위 벡터 \mathbf{u}:
\displaystyle |\mathbf{v}|=5,\mathbf{u}=\left\langle\dfrac{3}{5},\dfrac{4}{5}\right\rangle.
내적:
\displaystyle D_{\mathbf{u}}f=\langle 4,1\rangle\cdot\left\langle\dfrac{3}{5},\dfrac{4}{5}\right\rangle=4\left(\dfrac{3}{5}\right)+1\left(\dfrac{4}{5}\right)=\dfrac{12}{5}+\dfrac{4}{5}=\dfrac{16}{5}.