Fill in the blanks to complete each of the following theorem statements:

Let $f(x,y)$be a function of two variables and $\left(x_0,y_0\right)$be a point in the domain of $f$at which the first-order partial derivatives of $f$exist. If $u\in {\mathrm{ℝ}}^2$is a _____ for which the directional derivative $D_{u}f\left(x_0,y_0\right)$also exists, then $D_{u}f\left(x_0,y_0\right)=\_\_\_\_\_$.

Let $f(x,y)$be a function of two variables and $\left(x_0,y_0\right)$be a point in the domain of $f$at which the first-order partial derivatives of $f$exist. If $u\in {\mathrm{ℝ}}^2$is a _____ for which the directional derivative $D_{u}f\left(x_0,y_0\right)$also exists, then$D_{u}f\left(x_0,y_0\right)=\_\_\_\_\_$ .

Let $f(x,y)$be a function of two variables and $\left(x_0,y_0\right)$be a point in the domain of $f$at which the first-order partial derivatives of $f$exist. If $u\in {\mathrm{ℝ}}^2$ is a unit vectorfor which the directional derivative $D_{u}f\left(x_0,y_0\right)$also exists, then $D_{u}f\left(x_0,y_0\right)=\nabla f\left(x_0,y_0\right)·u$.

The rate of change of $f$in a different direction than usual positive $x$and $y$directions is the directional derivative of $f$in the direction of a specified unit vector $u$. The gradient provides a shortcut for finding the directional derivative when the first-order partial derivatives of the function exist.