https://file.aiquickdraw.com/mathbot/file/1726367034731jw4a3omi.jpg

We are tasked with finding the derivative \(\frac{dy}{dx}\) of the function \( y = (7x^2 + 10)\sin(2x) \).

### Step 1: Apply the Product Rule
The function is the product of two functions:
- \( u(x) = 7x^2 + 10 \)
- \( v(x) = \sin(2x) \)

The product rule states that:
\[
\frac{d}{dx}(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)
\]

### Step 2: Differentiate each part
- \( u(x) = 7x^2 + 10 \)
  \[
  u'(x) = \frac{d}{dx}(7x^2 + 10) = 14x
  \]
  
- \( v(x) = \sin(2x) \)
  \[
  v'(x) = \frac{d}{dx}(\sin(2x)) = 2\cos(2x) \quad (\text{using chain rule since the argument is } 2x)
  \]

### Step 3: Use the product rule formula
Now substitute into the product rule:
\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]
\[
\frac{dy}{dx} = (14x)\sin(2x) + (7x^2 + 10)(2\cos(2x))
\]

### Final Result
Thus, the derivative is:
\[
\frac{dy}{dx} = 14x\sin(2x) + (7x^2 + 10)2\cos(2x)
\]

This matches the **first option** in your image:
\[
14x\sin(2x) + (7x^2 + 10)2\cos(2x)
\]

Let me know if you'd like further details!

Here are some related questions:
1. What is the product rule and when do we use it in calculus?
2. How does the chain rule apply in differentiating trigonometric functions?
3. What are the steps for differentiating a product of two functions?
4. How do we differentiate composite functions like \(\sin(2x)\)?
5. Can we generalize this approach for products involving higher powers of \(x\)?

**Tip**: Always identify parts of a function that require the product or chain rule early in the differentiation process to avoid confusion.

Learn how to find the derivative dy/dx of the function y = (7x^2 + 10)sin(2x) using the product rule. This detailed solution breaks down the steps including the application of the product and chain rules. Suitable for undergraduate calculus students.

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