differentiate. y = ln (3x² - 5x + 7) y =

Explanation:

Step1: Identify the outer - inner functions

Let $u = 3x^{2}-5x + 7$, then $y=\ln(u)$.

Step2: Differentiate the outer function

The derivative of $y = \ln(u)$ with respect to $u$ is $\frac{dy}{du}=\frac{1}{u}$.

Step3: Differentiate the inner function

The derivative of $u = 3x^{2}-5x + 7$ with respect to $x$ is $\frac{du}{dx}=6x - 5$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{1}{u}$ and $\frac{du}{dx}=6x - 5$ into the chain - rule formula, and replace $u$ with $3x^{2}-5x + 7$. So $\frac{dy}{dx}=\frac{6x - 5}{3x^{2}-5x + 7}$.

Answer:

$\frac{6x - 5}{3x^{2}-5x + 7}$