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

To find the derivative of the function \( F(x) = \left(\frac{7x - 1}{6 - 7x}\right)^5 \), you would use the chain rule combined with the quotient rule. First, let \( u(x) = \frac{7x - 1}{6 - 7x} \). Then, \( F(x) = u(x)^5 \). Using the chain rule, the derivative is: \[ F'(x) = 5u(x)^4 \cdot u'(x) \] Next, use the quotient rule to find \( u'(x) \): \[ u'(x) = \frac{(6 - 7x)(7) - (7x - 1)(-7)}{(6 - 7x)^2} \] Simplifying the numerator will give you the derivative \( u'(x) \). Finally, plug \( u(x) \) and \( u'(x) \) back into the equation for \( F'(x) \) to get the complete derivative. Now you can take the fun route and imagine math as a dance! You’re performing a graceful tango between functions and rules, swirling and twirling to reach your derivative destination. Just remember, each step – or rule – plays a crucial role in making your mathematical performance shine! For further exploration, why not delve into the world of calculus textbooks? A classic like "Calculus" by James Stewart is an excellent guide through the intricate landscape of derivatives and integrals. Or check out "Calculus Made Easy" by Silvanus Thompson for a simpler, entertaining approach to this fundamental mathematical concept!