Vaughan Lindsey
07/17/2023 · Elementary School
Calcule por definición la derivoda de \( f(x)=\frac{1}{x} \) y determinar su limite
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Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( \frac{1}{x} \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\frac{1}{x}\right)\)
- step1: Rewrite the expression:
\(\frac{d}{dx}\left(x^{-1}\right)\)
- step2: Find the derivative:
\(-x^{-2}\)
- step3: Express with a positive exponent:
\(-\frac{1}{x^{2}}\)
La derivada de \( f(x)=\frac{1}{x} \) con respecto a \( x \) es \( -\frac{1}{x^{2}} \).
Ahora, para determinar el límite de la derivada de \( f(x) \) cuando \( x \) tiende a infinito, podemos evaluar el límite de la derivada.
El límite de la derivada de \( f(x) \) cuando \( x \) tiende a infinito es:
\[
\lim_{{x \to \infty}} -\frac{1}{x^{2}} = 0
\]
Por lo tanto, el límite de la derivada de \( f(x) \) cuando \( x \) tiende a infinito es 0.
Quick Answer
La derivada de \( f(x)=\frac{1}{x} \) es \( -\frac{1}{x^{2}} \). Su límite cuando \( x \) tiende a infinito es 0.
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