Directional Derivatives

Normal derivatives are taken with respect to one variable, i.e. going along its axis

Directional Derivatives are taken with respect to directions beside the standard horizontal and vertical axis

The direction will be defined by a vector

as there are infinite vectors, we default to an unit vector (length one for each direction

| | \tilde{u} | | = \sqrt{u_{1}^{2} + u_{2}^{2}} = 1

The slope of the tangent line to the intersection of a plane in the direction with the surface at $(x, y)$ is the directional derivative of $f (x, y)$ in the direction $\tilde{u}$

D_{\tilde{u}} f (x, y)

D_{\tilde{u}} f (a, b) = lim_{h \to 0} \frac{f (\tilde{a} + h \tilde{u})}{h} = lim_{h \to 0} \frac{f (a + h u_{1}, b + h u_{2}) - f (a, b)}{h}

If $f$ is a differentiable function of $x$ and $y$ then $f$ has a directional derivative in the direction of any unit vector $\tilde{u} = (u_{1}, u_{2})$ given by

D_{\tilde{u}} f (a, b) = f_{x} (a, b) \cdot u_{1} + f_{y} (a, b) \cdot u_{2}

Rewriting it using a dot product

D_{\tilde{u}} f (a, b) =< f_{x} (a, b), f_{y} (a, b) > \cdot < u_{1}, u_{2} >

The derivatives can be thought of as a Gradient Vector

D_{\tilde{u}} f (a, b) = \nabla f (a, b) \cdot \tilde{u}

Largest Derivative Direction

If $f$ has continuous first partials at $\tilde{a}$ and $\nabla f (\tilde{a}) \neq \tilde{0}$ then the largest value of the directional derivative $D_{\tilde{u}} f (\tilde{a})$ is $| | \nabla f (\tilde{a}) | |$ and it occurs when $\tilde{u}$ has the same direction as the gradient vector $\nabla f (\tilde{a})$

gradient always points in the direction of steepest ascent ( $- \nabla f$ points to steepest descent)

Level curves/surfaces

If $f (x, y)$ has continuous first partial derivatives at $(a, b)$ and $\nabla f (a, b) \neq \tilde{0}$ then $\nabla f (a, b)$ is orthogonal to the level curve $f (x, y) = k$ at $(a, b)$