e derivative of the function \( F(x)=\left(\frac{7 x-1}{6-7 x}\right)^{5} \)

To find the derivative of the function \( F(x)=\left(\frac{7 x-1}{6-7 x}\right)^{5} \), we can use the chain rule. Let \( u = \frac{7x-1}{6-7x} \), then \( F(x) = u^5 \). First, we need to find \( u' \), the derivative of \( u \). Using the quotient rule, we get: \[ u' = \frac{(7)(6-7x) - (7x-1)(-7)}{(6-7x)^2} = \frac{42 - 49x + 49x - 7}{(6-7x)^2} = \frac{35}{(6-7x)^2} \] Now we can apply the chain rule: \[ F'(x) = 5u^4 \cdot u' = 5 \left(\frac{7x-1}{6-7x}\right)^{4} \cdot \frac{35}{(6-7x)^2} \] So the final derivative \( F'(x) \) is: \[ F'(x) = \frac{175 \left(7x-1\right)^{4}}{\left(6-7x\right)^{6}} \]