Marsh Kelley
11/20/2023 · Middle School
Diferencia la siguiente función: \( y = \cos^{-1}(\frac{x}{2}) \)
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Tutor-Verified Answer
Step-by-step Solution
Para diferenciar la función \( y = \cos^{-1}\left(\frac{x}{2}\right) \), utilizamos la regla de la cadena. La derivada de \( \cos^{-1}(u) \) con respecto a \( u \) es \( -\frac{1}{\sqrt{1 - u^2}} \), y luego multiplicamos por la derivada de \( u \) con respecto a \( x \).
Definimos \( u = \frac{x}{2} \). Entonces, la derivada de \( u \) con respecto a \( x \) es:
\[
\frac{du}{dx} = \frac{1}{2}
\]
Ahora, aplicamos la regla de la cadena:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{1}{2}
\]
Sustituyendo \( u = \frac{x}{2} \):
\[
\frac{dy}{dx} = -\frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^2}} \cdot \frac{1}{2}
\]
Simplificando:
\[
\frac{dy}{dx} = -\frac{1}{2\sqrt{1 - \frac{x^2}{4}}}
\]
Finalmente, podemos escribir la derivada de la función \( y = \cos^{-1}\left(\frac{x}{2}\right) \) como:
\[
\frac{dy}{dx} = -\frac{1}{2\sqrt{1 - \frac{x^2}{4}}}
\]
Quick Answer
La derivada de \( y = \cos^{-1}\left(\frac{x}{2}\right) \) es \( \frac{dy}{dx} = -\frac{1}{2\sqrt{1 - \frac{x^2}{4}}} \).
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