Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.

The limit definition of the directional derivative of a function of two or three variables, $f$, at a point $P$in the direction of a unit vector $u$.

The limit definition of the directional derivative of a function of two or three variables, $f$, at a point $P$in the direction of a unit vector$u$.

Let $f(x,y)$be a function of two variables defined on an open set containing the point $(x_0,y_0)$, and let be a unit vector. The directional derivative of $f$at $(x_0,y_0)$in the direction of $u$, denoted by $D_{u}f(x_0,y_0)$, is given by $\underset{h\to 0}{\lim }\frac{f(x_0+\alpha h,y_0+\beta h)-f(x_0,y_0)}{h}$provided that this limit exists.

Example:

The directional derivative of the function, $f(x,y)=\frac{x^2}{y}$at the point $\left(-1,2\right)$in the direction of the vector role="math" localid="1654235626099" is calculated as below.

First, find the unit vector, $u=\frac{v}{||v||}$

The directional derivative, $D_{u}f(-1,2)=\underset{h\to 0}{\lim }\frac{{\displaystyle \frac{{\left(-1+{\displaystyle \frac{3h}{5}}\right)}^2}{2-{\displaystyle \frac{4h}{5}}}}-{\displaystyle \frac{{\left(-1\right)}^2}{2}}}{h}$

$$\begin{gathered} =\underset{h\to 0}{\lim }\frac{{\displaystyle \frac{{\displaystyle \frac{{\left(-5+3h\right)}^2}{25}}}{{\displaystyle \frac{10-4h}{5}}}}-{\displaystyle \frac{1}{2}}}{h} \\ =\underset{h\to 0}{\lim }\frac{2\left(9h^2+25-30h\right)-5\left(10-4h\right)}{10(10-4h)h} \\ =\underset{h\to 0}{\lim }\frac{18h^2+50-60h-50+20h}{10(10-4h)h} \\ =\underset{h\to 0}{\lim }\frac{18h^2-40h}{10(10-4h)h} \\ =\underset{h\to 0}{\lim }\frac{2h(9h-20)}{2h(50-20h)} \\ =\underset{h\to 0}{\lim }\frac{(9h-20)}{(50-20h)} \\ =\frac{9·0-20}{50-20·0} \\ =-\frac{20}{50} \\ =-\frac{2}{5} \end{gathered}$$