20. - / 1 points use logarithmic differentiation to find the derivative of the function. y = (sin(3x))^ln(x)…

Explanation:

Step1: Take natural - log of both sides

$\ln y=\ln((\sin(3x))^{\ln(x)})=\ln(x)\ln(\sin(3x))$

Step2: Differentiate both sides with respect to $x$

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = \ln(x)$ and $v=\ln(\sin(3x))$. The derivative of $\ln(x)$ is $\frac{1}{x}$, and for $\ln(\sin(3x))$, using the chain - rule, let $u=\sin(3x)$, then $\frac{d}{dx}\ln(\sin(3x))=\frac{1}{\sin(3x)}\cdot3\cos(3x)=3\cot(3x)$. So, $\frac{y^\prime}{y}=\frac{1}{x}\ln(\sin(3x))+\ln(x)\cdot3\cot(3x)$

Step3: Solve for $y^\prime$

Multiply both sides by $y = (\sin(3x))^{\ln(x)}$ $y^\prime=(\sin(3x))^{\ln(x)}\left(\frac{\ln(\sin(3x))}{x}+3\ln(x)\cot(3x)\right)$

Answer:

$(\sin(3x))^{\ln(x)}\left(\frac{\ln(\sin(3x))}{x}+3\ln(x)\cot(3x)\right)$