Compute the directional derivative of the following function at the given point in the direction of the given vector. Be
sure to use a unit vector for the direction vector.

The directional derivative is
(Type an exact answer, using radicals as needed.)

Ask by Lynch Reese. in the United States

Mar 17,2025

Question

Compute the directional derivative of the following function at the given point in the direction of the given vector. Be
sure to use a unit vector for the direction vector.

The directional derivative is
(Type an exact answer, using radicals as needed.)

Ask by Lynch Reese. in the United States

Mar 17,2025

UpStudy AI · Accepted Answer

We begin with the function

$$
f(x, y)=\sqrt{9-x^{2}-3y}=(9-x^{2}-3y)^{\frac{1}{2}}.
$$

**Step 1. Compute the gradient of $$ f $$.**

The partial derivative with respect to $$ x $$ is

$$
f_x(x,y) = \frac{1}{2}(9 - x^2 - 3y)^{-\frac{1}{2}} (-2x) = -\frac{x}{\sqrt{9-x^2-3y}}.
$$

The partial derivative with respect to $$ y $$ is

$$
f_y(x,y) = \frac{1}{2}(9 - x^2 - 3y)^{-\frac{1}{2}} (-3) = -\frac{3}{2\sqrt{9-x^2-3y}}.
$$

Thus, the gradient is

$$
\nabla f(x,y)= \left( -\frac{x}{\sqrt{9-x^2-3y}},\ -\frac{3}{2\sqrt{9-x^2-3y}} \right).
$$

**Step 2. Evaluate the gradient at $$ P(3,-3) $$.**

First, calculate the denominator at $$ P(3,-3) $$:

$$
9-x^2-3y \quad \text{at} \quad (3,-3)= 9-3^2-3(-3)= 9-9+9= 9.
$$

So, $$ \sqrt{9} = 3 $$.

Now, evaluate each component:

$$
f_x(3,-3)= -\frac{3}{3}= -1,
$$
$$
f_y(3,-3)= -\frac{3}{2\cdot 3}= -\frac{1}{2}.
$$

Thus,

$$
\nabla f(3,-3)=(-1,\ -\frac{1}{2}).
$$

**Step 3. Use the unit direction vector and compute the directional derivative.**

The unit direction vector is given as

$$
\mathbf{u} = \left(\frac{1}{\sqrt{10}},\ \frac{3}{\sqrt{10}}\right).
$$

The directional derivative $$ D_{\mathbf{u}}f $$ is the dot product of the gradient and the unit vector:

$$
D_{\mathbf{u}}f = \nabla f(3,-3) \cdot \mathbf{u}= (-1,\ -\frac{1}{2}) \cdot \left(\frac{1}{\sqrt{10}},\ \frac{3}{\sqrt{10}}\right).
$$

Calculate the dot product:

$$
D_{\mathbf{u}}f = -1 \cdot \frac{1}{\sqrt{10}} + \left(-\frac{1}{2}\right) \cdot \frac{3}{\sqrt{10}} = -\frac{1}{\sqrt{10}} - \frac{3}{2\sqrt{10}}.
$$

Combine the terms:

$$
-\frac{1}{\sqrt{10}} - \frac{3}{2\sqrt{10}} = -\frac{2}{2\sqrt{10}} - \frac{3}{2\sqrt{10}} = -\frac{5}{2\sqrt{10}}.
$$

**Final Answer:**

$$
-\frac{5}{2\sqrt{10}}.
$$
The directional derivative is $$ -\frac{5}{2\sqrt{10}} $$.

Compute the directional derivative of the following function at the given point in the direction of the given vector. Be
sure to use a unit vector for the direction vector.

The directional derivative is
(Type an exact answer, using radicals as needed.)

Upstudy AI Solution

Answer

Solution

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Compute the directional derivative of the following function at the given point in the direction of the given vector. Be sure to use a unit vector for the direction vector. The directional derivative is (Type an exact answer, using radicals as needed.)

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Compute the directional derivative of the following function at the given point in the direction of the given vector. Be
sure to use a unit vector for the direction vector.

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(Type an exact answer, using radicals as needed.)