evaluate and simplify y. y = csc^4(6x) y = □

Explanation:

Step1: Apply chain - rule

Let $u = \csc(6x)$, then $y = u^{4}$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. Using the power rule, if $y = u^{4}$, then $\frac{dy}{du}=4u^{3}$.

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

Let $v = 6x$, then $u=\csc(v)$. By the chain - rule $\frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}$. The derivative of $\csc(v)$ with respect to $v$ is $-\csc(v)\cot(v)$, and $\frac{dv}{dx}=6$. So $\frac{du}{dx}=- 6\csc(6x)\cot(6x)$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $u = \csc(6x)$ and $\frac{du}{dx}=-6\csc(6x)\cot(6x)$ into $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. We get $\frac{dy}{dx}=4u^{3}\cdot(-6\csc(6x)\cot(6x))$. Replace $u$ with $\csc(6x)$: $\frac{dy}{dx}=4\csc^{3}(6x)\cdot(-6\csc(6x)\cot(6x))$.

Step4: Simplify

Multiply the coefficients and combine the $\csc$ terms: $\frac{dy}{dx}=-24\csc^{4}(6x)\cot(6x)$.

Answer:

$-24\csc^{4}(6x)\cot(6x)$