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\( y = ( x ^ { 2 } - 1 ) ^ { \operatorname { Ln } ( x ^ { 2 } - 1 ) } \)

Ask by Fuentes Morrison. in Colombia
Mar 16,2025

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La derivada de \( y = (x^2 - 1)^{\ln(x^2 - 1)} \) es: \[ \frac{dy}{dx} = \frac{4x \ln(x^2 - 1)}{x^2 - 1} \cdot (x^2 - 1)^{\ln(x^2 - 1)} \]

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This fascinating function combines polynomial and logarithmic expressions, creating a unique landscape in mathematics! The expression \( (x^{2} - 1) \) can lead to different behaviors based on the value of \( x \). For instance, when \( x \) is between -1 and 1, \( (x^{2} - 1) \) becomes negative, making the logarithm undefined—be careful where you tread within those bounds! To solve or analyze this function effectively, consider its critical points and exponential nature. Plotting the graph helps visualize how the function behaves as it approaches undefined territories or asymptotes. One common mistake is attempting to evaluate the function without considering the domain. Always check the validity of your inputs to steer clear of mathematical pitfalls!

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