if y = 3xln(x) dy/dx = □

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$. Here, $u = 3x$ and $v=\ln(x)$.

Step2: Find $\frac{du}{dx}$ and $\frac{dv}{dx}$

The derivative of $u = 3x$ with respect to $x$ is $\frac{du}{dx}=3$ (using the power - rule $\frac{d}{dx}(ax)=a$ for a constant $a$). The derivative of $v=\ln(x)$ with respect to $x$ is $\frac{dv}{dx}=\frac{1}{x}$ (since $\frac{d}{dx}\ln(x)=\frac{1}{x}$).

Step3: Substitute into product - rule formula

$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}=3x\times\frac{1}{x}+\ln(x)\times3$.

Step4: Simplify the expression

$3x\times\frac{1}{x}=3$, so $\frac{dy}{dx}=3 + 3\ln(x)=3(1+\ln(x))$.

Answer:

$3(1 + \ln(x))$