Question

use logarithmic differentiation to find the derivative of the function. y = (cos(6x))^x y(x) = blank resources read it watch it submit answer 20. -/1 points use logarithmic differentiation to find the derivative of the function. y = (sin(3x))^ln(x) y(x) = blank resources read it

Explanation:

Step1: Take natural - log of both sides

Given $y = (\cos(6x))^{x}$, take $\ln$ of both sides: $\ln y=x\ln(\cos(6x))$.

Step2: Differentiate both sides with respect to $x$

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = x$ and $v=\ln(\cos(6x))$. The derivative of $\ln y$ with respect to $x$ is $\frac{y^\prime}{y}$, the derivative of $x$ is $1$, and for $\ln(\cos(6x))$, using the chain - rule, if $u=\cos(6x)$, then $\frac{d}{dx}\ln(u)=\frac{1}{u}\cdot\frac{du}{dx}$, and $\frac{du}{dx}=- 6\sin(6x)$. So $\frac{d}{dx}\ln(\cos(6x))=\frac{-6\sin(6x)}{\cos(6x)}=-6\tan(6x)$. Then $\frac{y^\prime}{y}=1\cdot\ln(\cos(6x))+x\cdot(-6\tan(6x))=\ln(\cos(6x)) - 6x\tan(6x)$.

Step3: Solve for $y^\prime$

Multiply both sides by $y = (\cos(6x))^{x}$ to get $y^\prime=(\cos(6x))^{x}(\ln(\cos(6x)) - 6x\tan(6x))$.

Answer:

$(\cos(6x))^{x}(\ln(\cos(6x)) - 6x\tan(6x))$