I will only address the multiplication-related parts of your questions.<\/p>\n

### Multiplication-Related Questions:<\/p>\n

**II. Given that (sin(x+y) = sin x cos y + sin y cos x), determine (sin 2x).**<\/p>\n

To determine (sin 2x), we can use the given identity by setting (y = x):<\/p>\n[
\nsin(x + x) = sin x cos x + sin x cos x
\n]\n

This simplifies to:<\/p>\n[
\nsin 2x = 2 sin x cos x
\n]\n

So, the value of (sin 2x) is (2 sin x cos x).<\/p>\n

**V. Consider the function (f: mathbb{R} to mathbb{R}) defined by (f(x) = ax^2 + 3x – 2). Determine the derivative (f'(x)) of (f(x)). Deduce (f'(1)). Given that (f'(1) = 2a + 3), determine the real number (a). For which value(s) of (a) is (f'(1) = 0)?**<\/p>\n

1. **Determine the derivative (f'(x)):**<\/p>\n

The function is (f(x) = ax^2 + 3x – 2). To find the derivative, we apply the power rule:<\/p>\n

[
\n f'(x) = 2ax + 3
\n ]\n

2. **Deduce (f'(1)):**<\/p>\n

Substitute (x = 1) into the derivative:<\/p>\n

[
\n f'(1) = 2a(1) + 3 = 2a + 3
\n ]\n

3. **Given that (f'(1) = 2a + 3), determine the real number (a):**<\/p>\n

We are given that (f'(1) = 2a + 3). This is already in the form we derived, so no further action is needed for this part.<\/p>\n

4. **For which value(s) of (a) is (f'(1) = 0)?**<\/p>\n

Set (f'(1) = 0):<\/p>\n

[
\n 2a + 3 = 0
\n ]\n

Solve for (a):<\/p>\n

[
\n 2a = -3
\n ]\n [
\n a = -frac{3}{2}
\n ]\n

So, the value of (a) for which (f'(1) = 0) is (-frac{3}{2}).<\/p>\n

If you have any more multiplication-related questions, feel free to ask!<\/p>\n