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b) \( g(x)=\frac{\operatorname{sen}(x)}{\cos (2 x)} \)

Ask by Cummings Phillips. in Chile
Jan 27,2025

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Answer

La función \( g(x) = \frac{\sin(x)}{\cos(2x)} \) tiene un dominio de todos los números reales excepto en \( x = \frac{\pi}{4} + \frac{k\pi}{2} \) para cualquier entero \( k \). La derivada de \( g(x) \) es \( g'(x) = \frac{\cos(x) \cos(2x) + 2\sin(x)\sin(2x)}{\cos^2(2x)} \).

Solution

Parece que hubo un problema al intentar calcular la derivada de la función \( g(x) = \frac{\sin(x)}{\cos(2x)} \). Sin embargo, puedo explicarte cómo calcularla manualmente. ### Cálculo de la Derivada Usamos la regla del cociente para derivar \( g(x) \): \[ g'(x) = \frac{u'v - uv'}{v^2} \] donde: - \( u = \sin(x) \) y \( u' = \cos(x) \) - \( v = \cos(2x) \) y \( v' = -2\sin(2x) \) Sustituyendo en la fórmula: \[ g'(x) = \frac{\cos(x) \cos(2x) - \sin(x)(-2\sin(2x))}{\cos^2(2x)} \] Simplificando: \[ g'(x) = \frac{\cos(x) \cos(2x) + 2\sin(x)\sin(2x)}{\cos^2(2x)} \] ### Resumen 1. **Dominio**: \( x \in \mathbb{R} \) excepto en \( x = \frac{\pi}{4} + \frac{k\pi}{2} \). 2. **Asíntotas verticales**: En los puntos donde \( \cos(2x) = 0 \). 3. **Derivada**: \( g'(x) = \frac{\cos(x) \cos(2x) + 2\sin(x)\sin(2x)}{\cos^2(2x)} \). Si necesitas más información o un análisis más profundo sobre la función, házmelo saber.

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The Deep Dive

Did you know that the function \( g(x) = \frac{\sin(x)}{\cos(2x)} \) ties back to some fascinating trigonometric identities? Specifically, using the double angle formula, we know that \( \cos(2x) = \cos^2(x) - \sin^2(x) \). This means that you can rewrite this function in diverse forms, deepening your understanding of its behavior over different intervals. When working with generic trigonometric functions, a common pitfall is overlooking the domain. In this case, \( g(x) \) will be undefined whenever \( \cos(2x) = 0 \), which happens at specific \( x \) values, such as \( x = \frac{\pi}{4} + \frac{k\pi}{2} \) for any integer \( k \). Always ensure to check for these restrictions to avoid calculation errors!

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