Questions: y=3 ln (x+8)/x^2 y'=

Solution Steps

To find the derivative of the function \( y = \frac{3 \ln(x+8)}{x^2} \), we will use the quotient rule. The quotient rule states that if you have a function \( y = \frac{u(x)}{v(x)} \), then its derivative is given by \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = 3 \ln(x+8) \) and \( v(x) = x^2 \). We will find the derivatives \( u'(x) \) and \( v'(x) \), and then apply the quotient rule.

We start with the function given by \[ y = \frac{3 \ln(x + 8)}{x^2} \] where \( u = 3 \ln(x + 8) \) and \( v = x^2 \).

Next, we compute the derivatives of \( u \) and \( v \): \[ u' = \frac{3}{x + 8} \] \[ v' = 2x \]

Using the quotient rule, we find the derivative \( y' \): \[ y' = \frac{u'v - uv'}{v^2} \] Substituting the values we calculated: \[ y' = \frac{\left(\frac{3}{x + 8}\right) x^2 - (3 \ln(x + 8))(2x)}{(x^2)^2} \]

This simplifies to: \[ y' = \frac{\frac{3x^2}{x + 8} - 6x \ln(x + 8)}{x^4} \]

Final Answer