b) \( y=\csc ^{5}\left(2 x+x^{3}\right)^{4} ; \frac{d}{d x} \)

To differentiate \( y = \csc^5\left(2x + x^3\right)^4 \) with respect to \( x \), you can use the chain rule and product rule. Start by setting \( u = \left(2x + x^3\right)^4 \). The derivative will involve \( \frac{dy}{du} \) and \( \frac{du}{dx} \). 1. Differentiate \( y \) using the chain rule: \[ \frac{dy}{dx} = 5\csc^4(u) \cdot (-\csc(u)\cot(u)) \cdot \frac{du}{dx} \] 2. Next, differentiate \( u \): \[ u = (2x + x^3)^4 \] \[ \frac{du}{dx} = 4(2x + x^3)^3 \cdot (2 + 3x^2) \] 3. Combine these results to find \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = 5\csc^4\left(2x + x^3\right)^4 \cdot (-\csc\left(2x + x^3\right)\cot\left(2x + x^3\right)) \cdot 4(2x + x^3)^3 \cdot (2 + 3x^2) \] Don’t forget to simplify where possible to get to your final answer!