https://file.aiquickdraw.com/mathbot/file/1726306456548y3d8yv9m.jpg

We are given a function \( y = (2x^2 + 13) \sin(5x) \), and the task is to find its derivative \( \frac{dy}{dx} \).

To solve this, we will apply the **product rule**, which states that if \( y = u(x) v(x) \), then:

\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]

Here, \( u(x) = 2x^2 + 13 \) and \( v(x) = \sin(5x) \).

### Step 1: Differentiate \( u(x) \)
\[
u'(x) = \frac{d}{dx}(2x^2 + 13) = 4x
\]

### Step 2: Differentiate \( v(x) \)
Since \( v(x) = \sin(5x) \), using the chain rule, we get:
\[
v'(x) = 5 \cos(5x)
\]

### Step 3: Apply the product rule
Now, applying the product rule:
\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]
Substitute \( u'(x) = 4x \), \( u(x) = 2x^2 + 13 \), \( v(x) = \sin(5x) \), and \( v'(x) = 5 \cos(5x) \):
\[
\frac{dy}{dx} = (4x)\sin(5x) + (2x^2 + 13)(5 \cos(5x))
\]

### Conclusion:
The derivative is:
\[
\frac{dy}{dx} = 4x \sin(5x) + 5(2x^2 + 13) \cos(5x)
\]

This matches the second option from your image, which is marked correctly.

Would you like more details on this solution or have any specific questions?

### Related Questions:
1. How would you apply the product rule if \( v(x) = e^{5x} \) instead of \( \sin(5x) \)?
2. Can you solve for the second derivative \( \frac{d^2 y}{dx^2} \) of this function?
3. How does the chain rule work when differentiating trigonometric functions?
4. What happens if the function was \( y = (3x^2 + 5x) \sin(4x) \)? Can you find the derivative?
5. How do you find the derivative of a product involving more than two functions?

### Tip:
Always remember to combine both the **product rule** and **chain rule** when differentiating composite functions like \( \sin(kx) \) where \( k \) is a constant.

Learn how to differentiate y = (2x^2 + 13)sin(5x) by applying the product rule and chain rule. The step-by-step solution explains how to find dy/dx, suitable for students in Grades 11-12 and early university calculus courses.

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