Question

Find the directional derivative of the function at the given point in the direction of the vector v. G(r, s) = tan−1(rs), (1, 3), v = 5i + 10j

Answer 1

The directional derivative of $f$ at the given point in the direction indicated is $\frac{5}{2}$.

How to calculate the directional derivative of a multivariate function

The directional derivative is represented by the following formula:

$\nabla_{\vec v} f = \nabla f (r_{o}, s_{o})\cdot \vec v$   (1)

Where:

• $\nabla f (r_{o}, s_{o})$ - Gradient evaluated at the point $(r_{o}, s_{o})$.
• $\vec v$ - Directional vector.

The gradient of $f$ is calculated below:

$\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{\partial f}{\partial r}(r_{o},s_{o}) \\\frac{\partial f}{\partial s}(r_{o},s_{o}) \end{array}\right]$   (2)

Where $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial s}$ are the partial derivatives with respect to $r$ and $s$, respectively.

If we know that $(r_{o}, s_{o}) = (1, 3)$, then the gradient is:

$\nabla f(r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{s}{1+r^{2}\cdot s^{2}} \\\frac{r}{1+r^{2}\cdot s^{2}}\end{array}\right]$

$\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{1+1^{2}\cdot 3^{2}} \\\frac{1}{1+1^{2}\cdot 3^{2}} \end{array}\right]$

$\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right]$

If we know that $\vec v = 5\,\hat{i} + 10\,\hat{j}$, then the directional derivative is:

$\nabla_{\vec v} f = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right] \cdot \left[\begin{array}{cc}5\\10\end{array}\right]$

$\nabla _{\vec v} f (r_{o}, s_{o}) = \frac{5}{2}$

The directional derivative of $f$ at the given point in the direction indicated is $\frac{5}{2}$. $\blacksquare$

To learn more on directional derivative, we kindly invite to check this verified question: brainly.com/question/9964491