find f(x). f(x)=9 ln (1 + 3x^2) f(x)=

Explanation:

Step1: Identify the outer - inner functions

Let $u = 1+3x^{2}$, then $y = 9\ln(u)$.

Step2: Differentiate the outer function with respect to $u$

The derivative of $y = 9\ln(u)$ with respect to $u$ is $\frac{dy}{du}=\frac{9}{u}$.

Step3: Differentiate the inner function with respect to $x$

The derivative of $u = 1 + 3x^{2}$ with respect to $x$ is $\frac{du}{dx}=6x$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{9}{u}$ and $\frac{du}{dx}=6x$ into the chain - rule formula. Since $u = 1+3x^{2}$, we have $\frac{dy}{dx}=\frac{9}{1 + 3x^{2}}\cdot6x$.

Step5: Simplify the expression

$\frac{dy}{dx}=\frac{54x}{1 + 3x^{2}}$.

Answer:

$\frac{54x}{1 + 3x^{2}}$