differentiate the following function.\ny = \\sqrt{\\ln 6x}\n\\frac{dy}{dx}=\\square\n(use parentheses to…

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt{\ln(6x)}$ as $y = (\ln(6x))^{\frac{1}{2}}$.

Step2: Apply the chain - rule

Let $u=\ln(6x)$. Then $y = u^{\frac{1}{2}}$. First, find $\frac{dy}{du}$ and $\frac{du}{dx}$. For $y = u^{\frac{1}{2}}$, by the power - rule $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}=\frac{1}{2\sqrt{u}}$. For $u=\ln(6x)$, by the chain - rule (since $\ln(ax)$ derivative is $\frac{a}{ax}=\frac{1}{x}$), $\frac{du}{dx}=\frac{6}{6x}=\frac{1}{x}$.

Step3: Use the chain - rule formula $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$

Substitute $u = \ln(6x)$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$. $\frac{dy}{dx}=\frac{1}{2\sqrt{\ln(6x)}}\cdot\frac{1}{x}=\frac{1}{2x\sqrt{\ln(6x)}}$.

Answer:

$\frac{1}{2x\sqrt{\ln(6x)}}$