Understand the directional derivative as the rate of change of \(f(x,y)\) in an arbitrary direction, generalizing the partial derivatives \(f_x\) and \(f_y\text{,}\) which are the special cases in the \(\mathbf i\) and \(\mathbf j\) directions.
Interpret the directional derivative geometrically as the slope of the tangent line to the curve in which a vertical plane through \(P_0\text{,}\) in the direction of \(\mathbf u\text{,}\) cuts the surface \(z = f(x,y)\text{.}\)
Understand the gradient vector \(\nabla f\) and its relationship to the directional derivative through \(\left(D_{\mathbf u} f\right)_{P_0}
= \left(\nabla f\right)_{P_0}\cdot\mathbf u\text{.}\)
Interpret the directional derivative through the angle between \(\nabla f\) and \(\mathbf u\text{,}\) and understand why \(f\) changes most rapidly in the direction of the gradient.
Recall that the partial derivatives \(f_x\) and \(f_y\) correspond to the rate of change of \(z = f(x,y)\) in the \(\mathbf i\) and \(\mathbf j\) directions. In this section, we generalize this notion to the rate of change of \(z = f(x,y)\) in an arbitrary direction \(\mathbf u\text{,}\) which is a unit vector of the form \(\mathbf u = u_1\mathbf i + u_2\mathbf j\text{.}\)
