Pregunta
Find the derivative \( y=\ln (3 x+1) \) Type answer \( y^{\prime}= \) (numerator)/(denominator) A
Ask by Luna Bartlett. in the United States
Mar 17,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( y' = \frac{3}{3x+1} \)
Solución
We start with
\[
y=\ln(3x+1)
\]
and differentiate using the chain rule.
1. The derivative of \(\ln(u)\) with respect to \(u\) is \(\frac{1}{u}\).
2. Here, \(u=3x+1\), and its derivative is
\[
\frac{du}{dx} = 3.
\]
3. By the chain rule,
\[
y'=\frac{du}{dx}\cdot \frac{1}{u}=\frac{3}{3x+1}.
\]
Thus, the derivative is:
\[
y'=\frac{3}{3x+1}.
\]
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Simplifique esta solución Beyond the Answer
To find the derivative of \( y = \ln(3x + 1) \), we can apply the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = 3x + 1 \). Thus, \( \frac{du}{dx} = 3 \). So, the derivative is: \[ y' = \frac{1}{3x + 1} \cdot 3 = \frac{3}{3x + 1} \] Putting this in the desired format: \( y' = 3/(3x + 1) \).
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