Weaver Savage
08/21/2024 · Elementary School
1. DE LA SIGUIENTE FUNCIÓN OBTÉN SU DERIVADA Derivada de \( f(x)=\left(5 x^{3}+2\right)^{-4} \)
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Para obtener la derivada de la función \( f(x) = (5x^3 + 2)^{-4} \), utilizaremos la regla de la cadena.
La regla de la cadena establece que si tenemos una función compuesta \( f(g(x)) \), su derivada es \( f'(g(x)) \cdot g'(x) \).
1. Identificamos las funciones:
- \( u = 5x^3 + 2 \)
- \( f(u) = u^{-4} \)
2. Calculamos la derivada de \( f(u) \):
\[
f'(u) = -4u^{-5}
\]
3. Calculamos la derivada de \( u \):
\[
u' = \frac{d}{dx}(5x^3 + 2) = 15x^2
\]
4. Aplicamos la regla de la cadena:
\[
f'(x) = f'(u) \cdot u' = -4(5x^3 + 2)^{-5} \cdot 15x^2
\]
5. Simplificamos:
\[
f'(x) = -60x^2(5x^3 + 2)^{-5}
\]
Por lo tanto, la derivada de \( f(x) = (5x^3 + 2)^{-4} \) es:
\[
f'(x) = -60x^2(5x^3 + 2)^{-5}
\]
Quick Answer
La derivada de \( f(x) = (5x^3 + 2)^{-4} \) es \( f'(x) = -60x^2(5x^3 + 2)^{-5} \).
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