Compute the gradient of the following function and evaluate it at the given point $$ P $$, $$ g(x, y)=x^{2}-8 x^{2} y-6 x y^{2} ; P(-1,3) $$ The gradient is $$
abla g(x, y)=\langle\square, \square $$.

Question

Compute the gradient of the following function and evaluate it at the given point $$ P $$, $$ g(x, y)=x^{2}-8 x^{2} y-6 x y^{2} ; P(-1,3) $$ The gradient is $$ 
abla g(x, y)=\langle\square, \square $$.

UpStudy AI · Accepted Answer

We start with the function
$$
g(x, y)=x^{2}-8x^{2}y-6xy^{2}.
$$

**Step 1. Compute the partial derivative with respect to $$ x $$:**

Differentiate each term treating $$ y $$ as a constant.

1. $$\frac{\partial}{\partial x}(x^2) = 2x$$.
2. $$\frac{\partial}{\partial x}(-8x^2y) = -8y \cdot 2x = -16xy$$.
3. $$\frac{\partial}{\partial x}(-6xy^2) = -6y^2$$.

Thus,
$$
\frac{\partial g}{\partial x} = 2x - 16xy - 6y^2.
$$

**Step 2. Compute the partial derivative with respect to $$ y $$:**

Differentiate each term treating $$ x $$ as a constant.

1. $$\frac{\partial}{\partial y}(x^2) = 0$$.
2. $$\frac{\partial}{\partial y}(-8x^2y) = -8x^2$$.
3. $$\frac{\partial}{\partial y}(-6xy^2) = -6x \cdot 2y = -12xy$$.

Thus,
$$
\frac{\partial g}{\partial y} = -8x^2 - 12xy.
$$

**Step 3. Write the gradient vector:**

The gradient is given by
$$
\nabla g(x, y)=\langle 2x - 16xy - 6y^2,\; -8x^2 - 12xy \rangle.
$$

**Step 4. Evaluate the gradient at $$ P(-1,3) $$:**

Substitute $$ x = -1 $$ and $$ y = 3 $$ into each component.

*For the $$ x $$-component:*
$$
2(-1) - 16(-1)(3) - 6(3)^2 = -2 + 48 - 54 = -8.
$$

*For the $$ y $$-component:*
$$
-8(-1)^2 - 12(-1)(3) = -8 + 36 = 28.
$$

Thus, at the point $$ P(-1,3) $$,
$$
\nabla g(-1,3)=\langle -8,\; 28 \rangle.
$$
The gradient of the function $$ g(x, y) = x^{2} - 8x^{2}y - 6xy^{2} $$ is $$ \nabla g(x, y) = \langle 2x - 16xy - 6y^{2}, -8x^{2} - 12xy \rangle $$. At the point $$ P(-1, 3) $$, the gradient is $$ \nabla g(-1, 3) = \langle -8, 28 \rangle $$.

Compute the gradient of the following function and evaluate it at the given point ,

The gradient is .

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