Problem 49
Question
Find the first partial derivatives of the following functions. $$h(w, x, y, z)=\frac{w z}{x y}$$
Step-by-Step Solution
Verified Answer
Answer: The first partial derivatives of the function h(w, x, y, z) are:
$$\frac{\partial h}{\partial w} = \frac{z}{x y}, \ \frac{\partial h}{\partial x} = \frac{-w z}{x^2 y}, \ \frac{\partial h}{\partial y} = \frac{-w z}{x y^2}, \ \frac{\partial h}{\partial z} = \frac{w}{x y}$$
1Step 1: Partial derivative with respect to w
To find the partial derivative of the function with respect to w, we differentiate h(w, x, y, z) with respect to w while treating x, y, and z as constants. Using the power rule, we get:
$$\frac{\partial h}{\partial w} = \frac{\partial}{\partial w}\left(\frac{w z}{x y}\right) = \frac{z}{x y}$$
2Step 2: Partial derivative with respect to x
Now let's find the partial derivative of the function with respect to x. We differentiate h(w, x, y, z) with respect to x while treating w, y, and z as constants. This time, we'll use the quotient rule:
$$\frac{\partial h}{\partial x} = \frac{\partial}{\partial x}\left(\frac{w z}{x y}\right) = \frac{-w z}{x^2 y}$$
3Step 3: Partial derivative with respect to y
Now we will find the partial derivative of the function with respect to y. Similarly, differentiate h(w, x, y, z) with respect to y while treating w, x, and z as constants. Using the quotient rule, we obtain:
$$\frac{\partial h}{\partial y} = \frac{\partial}{\partial y}\left(\frac{w z}{x y}\right) = \frac{-w z}{x y^2}$$
4Step 4: Partial derivative with respect to z
Finally, let's find the partial derivative of the function with respect to z. Differentiate h(w, x, y, z) with respect to z while treating w, x, and y as constants. Using the power rule, we get:
$$\frac{\partial h}{\partial z} = \frac{\partial}{\partial z}\left(\frac{w z}{x y}\right) = \frac{w}{x y}$$
In summary, the first partial derivatives of the function h(w, x, y, z) are:
$$\frac{\partial h}{\partial w} = \frac{z}{x y}, \ \frac{\partial h}{\partial x} = \frac{-w z}{x^2 y}, \ \frac{\partial h}{\partial y} = \frac{-w z}{x y^2}, \ \frac{\partial h}{\partial z} = \frac{w}{x y}$$
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