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Answer
La derivada de \( \operatorname{arcsec}(3x) \) con respecto a \( x \) es \( \frac{1}{x \sqrt{9x^2 - 1}} \).
Solution
Para derivar \( \operatorname{arcsec}(3x) \) con respecto a \( x \), aplicamos la regla de la cadena. La derivada de \( \operatorname{arcsec}(u) \) es:
\[
\frac{d}{dx} \left[ \operatorname{arcsec}(u) \right] = \frac{u'}{|u| \sqrt{u^2 - 1}}
\]
En este caso, \( u = 3x \), por lo que \( u' = 3 \). Sustituyendo en la fórmula:
\[
\frac{d}{dx} \left[ \operatorname{arcsec}(3x) \right] = \frac{3}{|3x| \sqrt{(3x)^2 - 1}}
\]
Simplificando:
\[
\frac{3}{|3x| \sqrt{9x^2 - 1}} = \frac{1}{x \sqrt{9x^2 - 1}}
\]
Por lo tanto, la derivada de \( \operatorname{arcsec}(3x) \) respecto a \( x \) es:
\[
\frac{d}{dx} \left[ \operatorname{arcsec}(3x) \right] = \frac{1}{x \sqrt{9x^2 - 1}}
\]
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Simplify this solution The Deep Dive
To differentiate \( \operatorname{arcsec}(3x) \), we can use the chain rule along with the derivative formula for arcsecant, which is given by: \[ \frac{d}{dx}[\operatorname{arcsec}(u)] = \frac{1}{|u| \sqrt{u^2 - 1}} \frac{du}{dx} \] In this case, \( u = 3x \), so we first find \( \frac{du}{dx} = 3 \). Now we can plug this into the formula: \[ \frac{d}{dx}[\operatorname{arcsec}(3x)] = \frac{1}{|3x| \sqrt{(3x)^2 - 1}} \cdot 3 \] This simplifies to: \[ \frac{3}{|3x| \sqrt{9x^2 - 1}} \] So, the derivative of \( \operatorname{arcsec}(3x) \) with respect to \( x \) is: \[ \frac{d[\operatorname{arcsec}(3x)]}{dx} = \frac{3}{|3x| \sqrt{9x^2 - 1}} \]
