differentiate. y = ln (4x² - 7x + 9) y = □

Explanation:

Step1: Apply chain - rule

Let $u = 4x^{2}-7x + 9$, then $y=\ln(u)$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Differentiate $y$ with respect to $u$

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

Step3: Differentiate $u$ with respect to $x$

The derivative of $u = 4x^{2}-7x + 9$ with respect to $x$ is $\frac{du}{dx}=8x - 7$.

Step4: Calculate $\frac{dy}{dx}$

By the chain - rule $\frac{dy}{dx}=\frac{1}{u}\cdot(8x - 7)$. Substitute $u = 4x^{2}-7x + 9$ back in, we get $\frac{dy}{dx}=\frac{8x - 7}{4x^{2}-7x + 9}$.

Answer:

$\frac{8x - 7}{4x^{2}-7x + 9}$