Question

Compute the first partial derivatives of the following functions. $$g(x, y, z)=\frac{4 x-2 y-2 z}{3 y-6 x-3 z}$$

Short 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 $$z$$: $$\frac{\partial g(x, y, z)}{\partial z} = \frac{12x - 12z}{(3y - 6x - 3z)^2}$$

Step-by-step solution

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: $$\frac{\partial g(x, y, z)}{\partial x} = \frac{24x}{(-6x + 3y - 3z)^2}$$

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: $$\frac{\partial g(x, y, z)}{\partial y} = \frac{12z}{(3y - 6x - 3z)^2}$$

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: $$\frac{\partial g(x, y, z)}{\partial x} = \frac{24x}{(-6x + 3y - 3z)^2}$$ $$\frac{\partial g(x, y, z)}{\partial y} = \frac{12z}{(3y - 6x - 3z)^2}$$ $$\frac{\partial g(x, y, z)}{\partial z} = \frac{12x - 12z}{(3y - 6x - 3z)^2}$$