Find `dy/dx if y= sin2x - x^3`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \sin(2x) - x^3\), we will differentiate each term of the function with respect to \(x\).

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = \sin(2x) - x^3 \]

2. **Differentiate \(y\) with respect to \(x\)**:
We will differentiate each term separately.

- **Differentiating \(\sin(2x)\)**:
- We use the chain rule here. The derivative of \(\sin(u)\) is \(\cos(u)\), and we also need to multiply by the derivative of the inner function \(u = 2x\).
- Therefore, the derivative of \(\sin(2x)\) is:
\[ \frac{d}{dx}[\sin(2x)] = \cos(2x) \cdot \frac{d}{dx}[2x] = \cos(2x) \cdot 2 = 2\cos(2x) \]

- **Differentiating \(-x^3\)**:
- We apply the power rule. The derivative of \(x^n\) is \(n x^{n-1}\).
- Thus, the derivative of \(-x^3\) is:
\[ \frac{d}{dx}[-x^3] = -3x^{3-1} = -3x^2 \]

3. **Combine the derivatives**:
Now we can combine the results from the differentiation:
\[ \frac{dy}{dx} = 2\cos(2x) - 3x^2 \]

### Final Answer:
\[ \frac{dy}{dx} = 2\cos(2x) - 3x^2 \]