Questions: Differentiate. y=ln(8x^2-9x+8) y'=

Solution Steps

To differentiate the given function \( y = \ln(8x^2 - 9x + 8) \), we will use the chain rule. The chain rule states that if you have a composite function \( y = \ln(u(x)) \), then the derivative \( y' \) is given by \( y' = \frac{1}{u(x)} \cdot u'(x) \). Here, \( u(x) = 8x^2 - 9x + 8 \), so we need to find \( u'(x) \) and then apply the chain rule.

1. Identify the inner function \( u(x) = 8x^2 - 9x + 8 \).
2. Differentiate \( u(x) \) to get \( u'(x) \).
3. Apply the chain rule to find \( y' \).

We start with the function given by \[ y = \ln(8x^2 - 9x + 8). \]

To differentiate \( y \), we apply the chain rule. The derivative of \( y \) is given by \[ y' = \frac{u'(x)}{u(x)}, \] where \( u(x) = 8x^2 - 9x + 8 \).

We find the derivative of \( u(x) \): \[ u'(x) = \frac{d}{dx}(8x^2 - 9x + 8) = 16x - 9. \]

Now we substitute \( u(x) \) and \( u'(x) \) into the derivative formula: \[ y' = \frac{16x - 9}{8x^2 - 9x + 8}. \]

Final Answer