The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{5}{2}[/tex].
The directional derivative is represented by the following formula:
[tex]\nabla_{\vec v} f = \nabla f (r_{o}, s_{o})\cdot \vec v[/tex] (1)
Where:
The gradient of [tex]f[/tex] is calculated below:
[tex]\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{\partial f}{\partial r}(r_{o},s_{o}) \\\frac{\partial f}{\partial s}(r_{o},s_{o}) \end{array}\right][/tex] (2)
Where [tex]\frac{\partial f}{\partial r}[/tex] and [tex]\frac{\partial f}{\partial s}[/tex] are the partial derivatives with respect to [tex]r[/tex] and [tex]s[/tex], respectively.
If we know that [tex](r_{o}, s_{o}) = (1, 3)[/tex], then the gradient is:
[tex]\nabla f(r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{s}{1+r^{2}\cdot s^{2}} \\\frac{r}{1+r^{2}\cdot s^{2}}\end{array}\right][/tex]
[tex]\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{1+1^{2}\cdot 3^{2}} \\\frac{1}{1+1^{2}\cdot 3^{2}} \end{array}\right][/tex]
[tex]\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right][/tex]
If we know that [tex]\vec v = 5\,\hat{i} + 10\,\hat{j}[/tex], then the directional derivative is:
[tex]\nabla_{\vec v} f = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right] \cdot \left[\begin{array}{cc}5\\10\end{array}\right][/tex]
[tex]\nabla _{\vec v} f (r_{o}, s_{o}) = \frac{5}{2}[/tex]
The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{5}{2}[/tex]. [tex]\blacksquare[/tex]
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