Problem 64
Question
Compute the first partial derivatives of the following functions. $$g(x, y, z)=\frac{4 x-2 y-2 z}{3 y-6 x-3 z}$$
Step-by-Step Solution
Verified Answer
Question: Compute the first partial derivatives of the function $$g(x, y, z) = \frac{4x - 2y - 2z}{3y - 6x - 3z}$$ with respect to each variable.
Answer: The first partial derivatives are:
1. With respect to $$x$$: $$\frac{\partial g(x, y, z)}{\partial x} = \frac{24x}{(-6x + 3y - 3z)^2}$$
2. With respect to $$y$$: $$\frac{\partial g(x, y, z)}{\partial y} = \frac{12z}{(3y - 6x - 3z)^2}$$
3. With respect to $$z$$: $$\frac{\partial g(x, y, z)}{\partial z} = \frac{12x - 12z}{(3y - 6x - 3z)^2}$$
1Step 1: Partial derivative with respect to x
To find the partial derivative of $$g(x, y, z)$$ with respect to $$x$$, we will treat $$y$$ and $$z$$ as constants. Now we can rewrite the function as: $$g(x) = \frac{4x - 2y - 2z}{-6x + 3y - 3z}$$.
Applying the quotient rule for the derivative, we get:
$$\frac{\partial g(x, y, z)}{\partial x} = \frac{(-6)(4 - 2y - 2z) - 4(-6x + 3y - 3z)}{(-6x + 3y - 3z)^2}=\frac{-24 + 12y + 12z + 24x - 12y + 12z}{(-6x + 3y - 3z)^2}$$
Now, we can simplify the expression:
$$\frac{\partial g(x, y, z)}{\partial x} = \frac{24x}{(-6x + 3y - 3z)^2}$$
2Step 2: Partial derivative with respect to y
To find the partial derivative of $$g(x, y, z)$$ with respect to $$y$$, we will treat $$x$$ and $$z$$ as constants. Now we can rewrite the function as: $$g(y) = \frac{4x - 2y - 2z}{3y - 6x - 3z}$$.
Applying the quotient rule for the derivative, we get:
$$\frac{\partial g(x, y, z)}{\partial y} = \frac{(-2)(3y - 6x -3z) - (4x - 2y - 2z)(3)}{(3y - 6x -3z)^2}=\frac{-6y + 12x + 6z - 12x + 6y + 6z}{(3y - 6x - 3z)^2}$$
Now, we can simplify the expression:
$$\frac{\partial g(x, y, z)}{\partial y} = \frac{12z}{(3y - 6x - 3z)^2}$$
3Step 3: Partial derivative with respect to z
To find the partial derivative of $$g(x, y, z)$$ with respect to $$z$$, we will treat $$x$$ and $$y$$ as constants. Now we can rewrite the function as: $$g(z) = \frac{4x - 2y - 2z}{3y - 6x - 3z}$$.
Applying the quotient rule for the derivative, we get:
$$\frac{\partial g(x, y, z)}{\partial z} = \frac{-2(3y - 6x - 3z) - (4x - 2y - 2z)(-3)}{(3y - 6x - 3z)^2}=\frac{-6y + 12x + 6z + 12x - 6y + 6z}{(3y - 6x - 3z)^2}$$
Now, we can simplify the expression:
$$\frac{\partial g(x, y, z)}{\partial z} = \frac{12x - 12z}{(3y - 6x - 3z)^2}$$
The first partial derivatives of the given function are:
$$\frac{\partial g(x, y, z)}{\partial x} = \frac{24x}{(-6x + 3y - 3z)^2}$$
$$\frac{\partial g(x, y, z)}{\partial y} = \frac{12z}{(3y - 6x - 3z)^2}$$
$$\frac{\partial g(x, y, z)}{\partial z} = \frac{12x - 12z}{(3y - 6x - 3z)^2}$$
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