Questions: y=ln sqrt(x+6) y'=

Solution Steps

To find the derivative of the function \( y = \ln \sqrt{x+6} \), we can use the chain rule. First, rewrite the function as \( y = \frac{1}{2} \ln(x+6) \). Then, differentiate using the chain rule: the derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u' \).

The given function is \( y = \ln \sqrt{x+6} \). We can rewrite this function using the property of logarithms: \( \ln \sqrt{u} = \frac{1}{2} \ln u \). Therefore, the function becomes: \[ y = \frac{1}{2} \ln(x+6) \]

To find the derivative \( y' \), we apply the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u' \). Here, \( u = x+6 \), so \( u' = 1 \). Therefore, the derivative is: \[ y' = \frac{1}{2} \cdot \frac{1}{x+6} \cdot 1 = \frac{1}{2(x+6)} \]

Final Answer