Scolon 3 "La Dervoda" Calcular las doribadas de las sig funciones matematieco obleniendo la regla baska realizando bs oporaciones funtamentako do manora de Aplical la regla de dorivadas. a) \( F(x)=\frac{\frac{6}{7} x^{3}-11 x^{2}+2 x^{-3}}{6 x^{2}} \)

Alright, I need to calculate the derivatives of the given mathematical functions using the basic operations and applying the derivative rules. The function provided is: \( F(x) = \frac{\frac{6}{7} x^{3} - 11 x^{2} + 2 x^{-3}}{6 x^{2}} \) First, I'll simplify the expression to make differentiation easier. Let's rewrite the function: \( F(x) = \frac{6}{7} x^{3} \cdot \frac{1}{6 x^{2}} - 11 x^{2} \cdot \frac{1}{6 x^{2}} + 2 x^{-3} \cdot \frac{1}{6 x^{2}} \) Simplifying each term: 1. \( \frac{6}{7} x^{3} \cdot \frac{1}{6 x^{2}} = \frac{6}{7} \cdot \frac{1}{6} \cdot x^{3-2} = \frac{1}{7} x \) 2. \( -11 x^{2} \cdot \frac{1}{6 x^{2}} = -\frac{11}{6} \) 3. \( 2 x^{-3} \cdot \frac{1}{6 x^{2}} = \frac{2}{6} x^{-3-2} = \frac{1}{3} x^{-5} \) So, the simplified function is: \( F(x) = \frac{1}{7} x - \frac{11}{6} + \frac{1}{3} x^{-5} \) Now, I'll find the derivative of each term with respect to \( x \): 1. \( \frac{d}{dx} \left( \frac{1}{7} x \right) = \frac{1}{7} \) 2. \( \frac{d}{dx} \left( -\frac{11}{6} \right) = 0 \) (since it's a constant) 3. \( \frac{d}{dx} \left( \frac{1}{3} x^{-5} \right) = \frac{1}{3} \cdot (-5) x^{-6} = -\frac{5}{3} x^{-6} \) Combining these derivatives, the derivative of \( F(x) \) is: \( F'(x) = \frac{1}{7} - \frac{5}{3} x^{-6} \) To express this in a more standard form, I'll write \( x^{-6} \) as \( \frac{1}{x^{6}} \): \( F'(x) = \frac{1}{7} - \frac{5}{3x^{6}} \) Alternatively, combining the terms over a common denominator: \( F'(x) = \frac{3x^{6} - 35}{21x^{6}} \) This is the derivative of the given function using basic differentiation rules.