To analyze the function \( f(x) = \frac{5x^2 + 3x - 2}{x^3 - 8} \), we can perform several tasks such as finding its domain, identifying any asymptotes, and calculating its derivative. Here, we'll focus on finding the derivative of the function using the quotient rule.
- Quotient Rule: The derivative of a quotient \(\frac{u}{v}\) is given by \(\frac{u'v - uv'}{v^2}\), where \(u = 5x^2 + 3x - 2\) and \(v = x^3 - 8\).
- Calculate Derivatives: Compute \(u'\) and \(v'\), the derivatives of the numerator and the denominator, respectively.
- Apply the Quotient Rule: Substitute \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula to find \(f'(x)\).
Dado \(u = 5x^2 + 3x - 2\) y \(v = x^3 - 8\), calculamos las derivadas:
\[
u' = \frac{d}{dx}(5x^2 + 3x - 2) = 10x + 3
\]
\[
v' = \frac{d}{dx}(x^3 - 8) = 3x^2
\]
La regla del cociente establece que la derivada de \(\frac{u}{v}\) es:
\[
f'(x) = \frac{u'v - uv'}{v^2}
\]
Sustituyendo los valores calculados:
\[
f'(x) = \frac{(10x + 3)(x^3 - 8) - (5x^2 + 3x - 2)(3x^2)}{(x^3 - 8)^2}
\]
La derivada de la función \(f(x) = \frac{5x^2 + 3x - 2}{x^3 - 8}\) es:
\[
\boxed{f'(x) = \frac{(10x + 3)(x^3 - 8) - (5x^2 + 3x - 2)(3x^2)}{(x^3 - 8)^2}}
\]
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