Find the first partial derivatives of the following function. \[ F(u, v, w)=\frac{u}{7 v+4 w} \] \[ \begin{array}{l} F_{u}= \\ F_{v}= \\ F_{w}= \end{array} \]

Step 1 ：The function given is \(F(u, v, w)=\frac{u}{7 v+4 w}\).

Step 2 ：We are asked to find the first partial derivatives of this function with respect to u, v, and w.

Step 3 ：Partial derivatives are a type of derivative where we differentiate with respect to one variable while keeping the other variables constant.

Step 4 ：The partial derivative of F with respect to u, denoted as \(F_u\), is the derivative of F with respect to u while treating v and w as constants.

Step 5 ：The partial derivative of F with respect to v, denoted as \(F_v\), is the derivative of F with respect to v while treating u and w as constants.

Step 6 ：The partial derivative of F with respect to w, denoted as \(F_w\), is the derivative of F with respect to w while treating u and v as constants.

Step 7 ：To find these derivatives, we can use the quotient rule for differentiation, which states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Step 8 ：Calculating these derivatives, we find that \(F_u = \frac{1}{7v + 4w}\), \(F_v = \frac{-7u}{(7v + 4w)^2}\), and \(F_w = \frac{-4u}{(7v + 4w)^2}\).

Step 9 ：\(\boxed{F_{u} = \frac{1}{7 v+4 w}}\)

Step 10 ：\(\boxed{F_{v} = \frac{-7 u}{(7 v+4 w)^{2}}}\)

Step 11 ：\(\boxed{F_{w} = \frac{-4 u}{(7 v+4 w)^{2}}}\)