`(e^(2x) + x^(3))/(cosec2x)`

Explanation

To find the derivative of the function \( y = \frac{e^{2x} + x^3}{\csc(2x)} \), we will follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function using the identity of cosecant:
\[ y = (e^{2x} + x^3) \cdot \sin(2x) \]

### Step 2: Apply the product rule
To differentiate \( y \), we will use the product rule. The product rule states that if \( y = u \cdot v \), then:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Here, let:
- \( u = e^{2x} + x^3 \)
- \( v = \sin(2x) \)

### Step 3: Differentiate \( u \) and \( v \)
Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. Differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(e^{2x}) + \frac{d}{dx}(x^3) = 2e^{2x} + 3x^2 \]

2. Differentiate \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx}(\sin(2x)) = 2\cos(2x) \]

### Step 4: Substitute into the product rule
Now substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the product rule:
\[ \frac{dy}{dx} = (e^{2x} + x^3)(2\cos(2x)) + \sin(2x)(2e^{2x} + 3x^2) \]

### Step 5: Simplify the expression
Now we can simplify the expression:
\[ \frac{dy}{dx} = 2(e^{2x} + x^3)\cos(2x) + \sin(2x)(2e^{2x} + 3x^2) \]

This is the final expression for the derivative \( \frac{dy}{dx} \).

### Final Answer
\[ \frac{dy}{dx} = 2(e^{2x} + x^3)\cos(2x) + \sin(2x)(2e^{2x} + 3x^2) \]

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