Q. 25 - Calculus

Question

In Exercises $21-28$, find the directional derivative of the given function at the specified point $\mathrm{P}$ and in the direction of the given unit vector$\mathbf{u}$ .

 $\mathrm{f}(\mathrm{x},\mathrm{y})=\sqrt{\frac{\mathrm{y}}{\mathrm{x}}}$at $\mathrm{P}=(4,9),\mathbf{u}=\left(-\frac{\sqrt{17}}{17},-\frac{4\sqrt{17}}{17}\right)$

Step-by-Step Solution

Answer

Directional derivative for the given function is $\nabla \mathrm{f}\left(\mathrm{P}\right)\times \mathrm{u}=-\frac{7}{816}\sqrt{17}$.

Step 1: Given information

Directional derivative of function is,

$\mathrm{f}(\mathrm{x},\mathrm{y})=\sqrt{\frac{\mathrm{y}}{\mathrm{x}}}$

Given is,

 $\mathrm{P}=\left({\mathrm{x}}_0,{\mathrm{y}}_0\right)=(4,9)$ and $\mathrm{u}=(\mathrm{\alpha },\mathrm{\beta })=\left(-\frac{\sqrt{17}}{17},-\frac{4\sqrt{17}}{17}\right)$

Step 2: Calculation for directional derivative

Direction derivative of function,

$\nabla \mathrm{f}\left(\mathrm{P}\right)\times \mathrm{u}=\nabla \mathrm{f}(4,9)\times \mathrm{u}$

$=\left({\left.\frac{\mathrm{df}}{\mathrm{dx}}\right|}_{(4,9)}\mathrm{i}+{\left.\frac{\mathrm{df}}{\mathrm{dy}}\right|}_{(4,9)}\mathrm{j}\right)\times \left(-\frac{\sqrt{17}}{17}\mathrm{i}-\frac{4\sqrt{17}}{17}\mathrm{j}\right)$

$=\left[{\left(\frac{1}{2\sqrt{\frac{\mathrm{y}}{\mathrm{x}}}}\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\right)}_{(4,9)}\mathrm{i}+{\left(\frac{1}{2\sqrt{\frac{\mathrm{y}}{\mathrm{x}}}}\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\right)}_{(4,9)}\mathrm{j}\right]\times \left(-\frac{\sqrt{17}}{17}\mathrm{i}-\frac{4\sqrt{17}}{17}\mathrm{j}\right)$

$=\left[{\left(\frac{\frac{-\mathrm{y}}{{\mathrm{x}}^2}}{2\sqrt{\frac{\mathrm{y}}{\mathrm{x}}}}\right)}_{(4,9)}\mathrm{i}+{\left(\frac{\frac{1}{\mathrm{x}}}{2\sqrt{\frac{\mathrm{y}}{\mathrm{x}}}}\right)}_{(4,9)}\mathrm{j}\right]\times \left(-\frac{\sqrt{17}}{17}\mathrm{i}-\frac{4\sqrt{17}}{17}\mathrm{j}\right)$

$=\left[\left(\frac{\frac{-9}{4^2}}{2\sqrt{\frac{9}{4}}}\right)\mathrm{i}+\left(\frac{\frac{1}{4}}{2\sqrt{\frac{9}{4}}}\right)\mathrm{j}\right]\times \left(-\frac{\sqrt{17}}{17}\mathrm{i}-\frac{4\sqrt{17}}{17}\mathrm{j}\right)$

$=\left[\left(\frac{-\frac{9}{16}}{2\times \frac{3}{2}}\right)\mathrm{i}+\left(\frac{\frac{1}{4}}{2\times \frac{3}{2}}\right)\mathrm{j}\right]\times \left(-\frac{\sqrt{17}}{17}\mathrm{i}-\frac{4\sqrt{17}}{17}\mathrm{j}\right)$

$=\left(-\frac{9}{48}\mathrm{i}+\frac{1}{12}\mathrm{j}\right)\times \left(-\frac{\sqrt{17}}{17}\mathrm{i}-\frac{4\sqrt{17}}{17}\mathrm{j}\right)$

$=\left(\frac{9\times \sqrt{17}}{48\times 17}-\frac{4\sqrt{17}}{12\times 17}\right)$

$=\left(\frac{9\times \sqrt{17}}{16\times 3\times 17}-\frac{\sqrt{17}}{3\times 17}\right)$

$=\frac{1}{3\times 17}\left(\frac{9\times \sqrt{17}}{16}-\frac{\sqrt{17}}{1}\right)$

$=\frac{1}{3\times 17}\left(\frac{9\times \sqrt{17}-16\sqrt{17}}{16}\right)$

$\nabla \mathrm{f}\left(\mathrm{P}\right)\times \mathrm{u}=-\frac{7}{816}\sqrt{17}$