question find the derivative of the function y = (2x√(3 - 3x))/(e^(4x)(sin²(x))) using logarithmic…

Explanation:

Step1: Take natural - log of both sides

$\ln y=\ln\left(\frac{2x\sqrt{3 - 3x}}{e^{4x}(\sin^{2}x)}\right)$ Using the properties of logarithms $\ln\left(\frac{a}{b}\right)=\ln a-\ln b$, $\ln(ab)=\ln a+\ln b$ and $\ln(a^{n})=n\ln a$, we get: $\ln y=\ln(2x)+\frac{1}{2}\ln(3 - 3x)-4x - 2\ln(\sin x)$

Step2: Differentiate both sides with respect to $x$

The derivative of $\ln y$ with respect to $x$ is $\frac{y'}{y}$ by the chain - rule. The derivative of $\ln(2x)$ is $\frac{1}{x}$, the derivative of $\frac{1}{2}\ln(3 - 3x)$ is $\frac{1}{2}\cdot\frac{-3}{3 - 3x}=\frac{-1}{2(1 - x)}$, the derivative of $-4x$ is $-4$, and the derivative of $-2\ln(\sin x)$ is $-2\cdot\frac{\cos x}{\sin x}=-2\cot x$. So, $\frac{y'}{y}=\frac{1}{x}-\frac{1}{2(1 - x)}-4 - 2\cot x$

Step3: Solve for $y'$

Multiply both sides by $y=\frac{2x\sqrt{3 - 3x}}{e^{4x}(\sin^{2}x)}$: $y'=\frac{2x\sqrt{3 - 3x}}{e^{4x}(\sin^{2}x)}\left(\frac{1}{x}-\frac{1}{2(1 - x)}-4 - 2\cot x\right)$

Answer:

$y'=\frac{2x\sqrt{3 - 3x}}{e^{4x}(\sin^{2}x)}\left(\frac{1}{x}-\frac{1}{2(1 - x)}-4 - 2\cot x\right)$