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Chapter 12: Problem 49

Find the first partial derivatives of the following functions. $$F(u, v, w)=\frac{u}{v+w}$$

Short Answer

Expert verified
Question: Determine the first partial derivatives for the function F(u, v, w) = u / (v + w). Solution: The first partial derivatives of F(u, v, w) are: 1) 鈭侳/鈭倁 = 1 / (v + w) 2) 鈭侳/鈭倂 = -u / (v + w)^2 3) 鈭侳/鈭倃 = -u / (v + w)^2

Step by step solution

01

Find the partial derivative with respect to u

To find the partial derivative of F(u, v, w) with respect to u, we will treat v and w as constants and differentiate F(u, v, w) with respect to u. Using the rules of differentiation, we have: $$\frac{\partial F}{\partial u} = \frac{\partial}{\partial u}\Big(\frac{u}{v+w}\Big)$$ Since v and w are constants, we can use the rule for differentiating a fraction: $$\frac{\partial F}{\partial u} = \frac{(v+w)\cdot 1 - u\cdot 0}{(v+w)^2} = \frac{1}{v+w}$$
02

Find the partial derivative with respect to v

Next, we will find the partial derivative of F(u, v, w) with respect to v. In this case, we will treat u and w as constants and differentiate F(u, v, w) with respect to v: $$\frac{\partial F}{\partial v} = \frac{\partial}{\partial v}\Big(\frac{u}{v+w}\Big)$$ Now, since u is constant, we can use the rule for differentiating a fraction: $$\frac{\partial F}{\partial v} = \frac{u \cdot 0 - u\cdot 1}{(v+w)^2} = -\frac{u}{(v+w)^2}$$
03

Find the partial derivative with respect to w

Finally, we will find the partial derivative of F(u, v, w) with respect to w. Here, we treat u and v as constants and differentiate F(u, v, w) with respect to w: $$\frac{\partial F}{\partial w} = \frac{\partial}{\partial w}\Big(\frac{u}{v+w}\Big)$$ Since u and v are constants, we can use the rule for differentiating a fraction: $$\frac{\partial F}{\partial w} = \frac{u \cdot 0 - u\cdot 1}{(v+w)^2} = -\frac{u}{(v+w)^2}$$ The first partial derivatives of the function F(u, v, w) are: $$\frac{\partial F}{\partial u} = \frac{1}{v+w}$$ $$\frac{\partial F}{\partial v} = -\frac{u}{(v+w)^2}$$ $$\frac{\partial F}{\partial w} = -\frac{u}{(v+w)^2}$$

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