Study guide and
4 practice problems
on:
Directional derivative definition
$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
The directional derivative of a function $f(x,y)$ is the rate of change of $f$ if $(x,y)$ is changed in the direction of $\bfv$.
Let $\bfx = (x,y)$. Then, $$D_\bfv f(\bfx) = \lim_{\epsilon \to 0} \frac{f(\bfx + \epsilon \bfv/|\bfv|) - f(\bfx)} {\epsilon} $$
Related topics
Directional derivative
(10 problems)
Functions of several variables
(36 problems)
Multivariable calculus
(147 problems)
Practice problems
Let $f(x,y) = x^2 + y^2$.
Describe the shape of the $f(x,y)=2$ level curve.
Without calculation, find the directional derivative at $(1,1)$ in the direction $-\bfi+\bfj$.
Hint: consider the level curve at $(1,1).$
By computation, find the directional derivative at $(1,1)$ in the direction of $-\bfi + \bfj$.
Solution
Here are equispaced level curves of a function $f(x,y)$.
a) Where is $\nabla f$ biggest in magnitude?
b) Where is $\nabla f$ smallest in magnitude?
c) Where is $\partial_x f =0$?
d) Where is the directional derivative $D_{(\mathbf{i}/\sqrt{2} + \mathbf{j}/\sqrt{2})} f = 0$?
Solution
Consider the surface given by $z = x^2 + y^2$. What is the (3d) tangent vector at $(1,1)$ that has an $\mathbf{i}$ component of 0 and a $\mathbf{j}$ component of 2?
Solution
Let $f(x,y) = x^2 + y^3 $ and $P=(1,1)$.
Find a direction at $P$ along which $f$ is not changing.
Solution