find f(x). f(x)=6 ln (3 + 5x^2) f(x)=

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $y = f(g(x))$, then $y'=f'(g(x))\cdot g'(x)$. Let $u = 3 + 5x^{2}$, so $y = 6\ln(u)$.

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

The derivative of $\ln(u)$ is $\frac{1}{u}$, so $\frac{dy}{du}=\frac{6}{u}$.

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

Differentiating $u = 3+5x^{2}$ with respect to $x$, we get $\frac{du}{dx}=10x$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{6}{u}$ and $\frac{du}{dx}=10x$ into the chain - rule formula. Since $u = 3 + 5x^{2}$, we have $\frac{dy}{dx}=\frac{6}{3 + 5x^{2}}\cdot10x$.

Step5: Simplify the expression

$\frac{6\times10x}{3 + 5x^{2}}=\frac{60x}{3 + 5x^{2}}$.

Answer:

$\frac{60x}{3 + 5x^{2}}$