Problem 45
Question
Find the first partial derivatives of the following functions. $$F(u, v, w)=\frac{u}{v+w}$$
Step-by-Step Solution
Verified Answer
Question: Find the first partial derivatives of the function F(u, v, w) = u / (v + w).
Answer: The first partial derivatives of the function F(u, v, w) with respect to u, v, and w are:
$$\frac{\partial F}{\partial u}=\frac{1}{v+w}$$
$$\frac{\partial F}{\partial v}=-\frac{u}{(v+w)^2}$$
$$\frac{\partial F}{\partial w}=-\frac{u}{(v+w)^2}$$
1Step 1: Calculate the Partial Derivative with respect to u
To find the partial derivative of the function F(u,v,w) with respect to u, we will treat v and w as constants and differentiate F with respect to u. The partial derivative with respect to u is denoted as:
$$\frac{\partial F}{\partial u}$$
The function we have is:
$$F(u, v, w)=\frac{u}{v+w}$$
Applying the derivative with respect to u, we have:
$$\frac{\partial F}{\partial u}=\frac{1}{v+w}$$
2Step 2: Calculate the Partial Derivative with respect to v
Now, we will find the partial derivative of the function F(u,v,w) with respect to v by treating u and w as constants. The partial derivative with respect to v is denoted as:
$$\frac{\partial F}{\partial v}$$
The function is still:
$$F(u, v, w)=\frac{u}{v+w}$$
Applying the derivative with respect to v, we have:
$$\frac{\partial F}{\partial v}=-\frac{u}{(v+w)^2}$$
3Step 3: Calculate the Partial Derivative with respect to w
Lastly, we will find the partial derivative of the function F(u,v,w) with respect to w, while treating u and v as constants. The partial derivative with respect to w is denoted as:
$$\frac{\partial F}{\partial w}$$
The function remains:
$$F(u, v, w)=\frac{u}{v+w}$$
Applying the derivative with respect to w, we have:
$$\frac{\partial F}{\partial w}=-\frac{u}{(v+w)^2}$$
4Step 4: Final Result
The first partial derivatives of the function F(u, v, w) with respect to u, v, and w are as follows:
$$\frac{\partial F}{\partial u}=\frac{1}{v+w}$$
$$\frac{\partial F}{\partial v}=-\frac{u}{(v+w)^2}$$
$$\frac{\partial F}{\partial w}=-\frac{u}{(v+w)^2}$$
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