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Chapter 12: Problem 54

Find the first partial derivatives of the following functions. $$F(w, x, y, z)=w \sqrt{x+2 y+3 z}$$

Short Answer

Expert verified
Answer: The first partial derivatives of the function F(w, x, y, z) are: 1. \(\frac{\partial F}{\partial w} = \sqrt{x+2 y+3 z}\) 2. \(\frac{\partial F}{\partial x} = \frac{w}{2\sqrt{x+2 y+3 z}}\) 3. \(\frac{\partial F}{\partial y} = \frac{2w}{2\sqrt{x+2 y+3 z}}\) 4. \(\frac{\partial F}{\partial z} = \frac{3w}{2\sqrt{x+2 y+3 z}}\)

Step by step solution

01

Finding the partial derivative with respect to w

To find the partial derivative concerning w, we need to treat x, y, and z as constants and differentiate the function with respect to w: $$\frac{\partial F}{\partial w} = \frac{\partial}{\partial w}(w \sqrt{x+2 y+3 z})$$ We can simplify this by using the rule for differentiating a product: $$\frac{\partial F}{\partial w} = \sqrt{x+2 y+3 z}$$
02

Finding the partial derivative with respect to x

To find the partial derivative concerning x, we need to treat w, y, and z as constants and differentiate the function with respect to x: $$\frac{\partial F}{\partial x} = \frac{\partial}{\partial x}(w \sqrt{x+2 y+3 z})$$ We can simplify this using the chain rule. First, differentiate the square root function and then differentiate the inner function with respect to x: $$\frac{\partial F}{\partial x} = w\frac{1}{2\sqrt{x+2 y+3 z}}\cdot\frac{\partial}{\partial x}(x+2 y+3 z) = \frac{w}{2\sqrt{x+2 y+3 z}}$$
03

Finding the partial derivative with respect to y

To find the partial derivative concerning y, we need to treat w, x, and z as constants and differentiate the function with respect to y: $$\frac{\partial F}{\partial y} = \frac{\partial}{\partial y}(w \sqrt{x+2 y+3 z})$$ Following the same process as before, we use the chain rule to differentiate the square root function and then differentiate the inner function concerning y: $$\frac{\partial F}{\partial y} = w\frac{1}{2\sqrt{x+2 y+3 z}}\cdot \frac{\partial}{\partial y}(x+2 y+3 z)= \frac{2w}{2\sqrt{x+2 y+3 z}}$$
04

Finding the partial derivative with respect to z

To find the partial derivative concerning z, we need to treat w, x, and y as constants and differentiate the function with respect to z: $$\frac{\partial F}{\partial z} = \frac{\partial}{\partial z}(w \sqrt{x+2 y+3 z})$$ Using the chain rule again to differentiate the square root function and then differentiate the inner function concerning z: $$\frac{\partial F}{\partial z} = w\frac{1}{2\sqrt{x+2 y+3 z}}\cdot \frac{\partial}{\partial z}(x+2 y+3 z) = \frac{3w}{2\sqrt{x+2 y+3 z}}$$ So, the first partial derivatives of the function F(w, x, y, z) are: 1. \(\frac{\partial F}{\partial w} = \sqrt{x+2 y+3 z}\) 2. \(\frac{\partial F}{\partial x} = \frac{w}{2\sqrt{x+2 y+3 z}}\) 3. \(\frac{\partial F}{\partial y} = \frac{2w}{2\sqrt{x+2 y+3 z}}\) 4. \(\frac{\partial F}{\partial z} = \frac{3w}{2\sqrt{x+2 y+3 z}}\)

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