Questions: For the functions w=xy+yz+xz, x=3u+v, y=3u-v, and z=uv, express the partial derivative of w with respect to u and the partial derivative of w with respect to v using the chain rule and by expressing w directly in terms of u and v before differentiating. Then evaluate the partial derivative of w with respect to u and the partial derivative of w with respect to v at the point (u, v)=(-1/4, 3). Express the partial derivative of w with respect to u and the partial derivative of w with respect to v as functions of u and v.

For the functions w=xy+yz+xz, x=3u+v, y=3u-v, and z=uv, express the partial derivative of w with respect to u and the partial derivative of w with respect to v using the chain rule and by expressing w directly in terms of u and v before differentiating. Then evaluate the partial derivative of w with respect to u and the partial derivative of w with respect to v at the point (u, v)=(-1/4, 3).

Express the partial derivative of w with respect to u and the partial derivative of w with respect to v as functions of u and v.

Transcript text: For the functions $w=x y+y z+x z, x=3 u+v, y=3 u-v$, and $z=u v$, express $\frac{\partial w}{\partial u}$ and $\frac{\partial w}{\partial v}$ using the chain rule and by expressing $w$ directly in terms of $u$ and $v$ before differentiating. Then evaluate $\frac{\partial w}{\partial u}$ and $\frac{\partial \mathrm{w}}{\partial \mathrm{v}}$ at the point $(u, v)=\left(-\frac{1}{4}, 3\right)$. Express $\frac{\partial w}{\partial u}$ and $\frac{\partial w}{\partial v}$ as functions of $u$ and $v$.

Solution

Solution Steps

To solve this problem, we need to use the chain rule to express the partial derivatives of $ w $ with respect to $ u $ and $ v $. First, substitute the expressions for $ x $, $ y $, and $ z $ in terms of $ u $ and $ v $ into the function $ w $. Then, differentiate the resulting expression for $ w $ with respect to $ u $ and $ v $. Finally, evaluate these derivatives at the given point $(u, v) = \left(-\frac{1}{4}, 3\right)$.

Step 1: Express $ w $ in Terms of $ u $ and $ v $

Given the functions:

$ x = 3u + v $
$ y = 3u - v $
$ z = uv $

The function $ w $ is defined as: \[ w = xy + yz + xz \]

Substituting the expressions for $ x $, $ y $, and $ z $ into $ w $: \[ w = (3u + v)(3u - v) + (3u - v)(uv) + (3u + v)(uv) \]

Simplifying: \[ w = (9u^2 - v^2) + (3u^2v - uv^2) + (3u^2v + uv^2) \] \[ w = 9u^2 - v^2 + 6u^2v \]

Step 2: Differentiate $ w $ with Respect to $ u $

Calculate the partial derivative of $ w $ with respect to $ u $: \[ \frac{\partial w}{\partial u} = \frac{\partial}{\partial u}(9u^2 - v^2 + 6u^2v) \] \[ = 18u + 12uv \]

Step 3: Differentiate $ w $ with Respect to $ v $

Calculate the partial derivative of $ w $ with respect to $ v $: \[ \frac{\partial w}{\partial v} = \frac{\partial}{\partial v}(9u^2 - v^2 + 6u^2v) \] \[ = -2v + 6u^2 \]

Step 4: Evaluate the Derivatives at $(u, v) = \left(-\frac{1}{4}, 3\right)$

Substitute $ u = -\frac{1}{4} $ and $ v = 3 $ into the partial derivatives:

For $ \frac{\partial w}{\partial u} $: \[ \frac{\partial w}{\partial u} = 18\left(-\frac{1}{4}\right) + 12\left(-\frac{1}{4}\right)(3) \] \[ = -\frac{9}{2} - 9 \] \[ = -\frac{27}{2} \] \[ = -13.5 \]

For $ \frac{\partial w}{\partial v} $: \[ \frac{\partial w}{\partial v} = -2(3) + 6\left(-\frac{1}{4}\right)^2 \] \[ = -6 + \frac{3}{2} \] \[ = -\frac{12}{2} + \frac{3}{2} \] \[ = -\frac{9}{2} \] \[ = -4.5 \]

Final Answer

The partial derivatives evaluated at $(u, v) = \left(-\frac{1}{4}, 3\right)$ are: \[ \boxed{\frac{\partial w}{\partial u} = -13.5} \] \[ \boxed{\frac{\partial w}{\partial v} = -4.5} \]

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