`"Let "f(x)=|{:(cos(x+x^(2)),sin (x+x^(2)),-cos(x+x^(2))),(sin (x-x^(2)),cos (x-x^(2)),sin (x-x^(2))),(sin 2x, 0, sin (2x^(2))):}|.`
Find the value of f'(0).

Explanation

To find the value of \( f'(0) \) for the given function

\[ f(x) = \begin{vmatrix} \cos(x + x^2) & \sin(x + x^2) & -\cos(x + x^2) \\ \sin(x - x^2) & \cos(x - x^2) & \sin(x - x^2) \\ \sin(2x) & 0 & \sin(2x^2) \end{vmatrix} \]

we will follow these steps:

### Step 1: Evaluate \( f(0) \)

Substituting \( x = 0 \):

\[ f(0) = \begin{vmatrix} \cos(0 + 0^2) & \sin(0 + 0^2) & -\cos(0 + 0^2) \\ \sin(0 - 0^2) & \cos(0 - 0^2) & \sin(0 - 0^2) \\ \sin(2 \cdot 0) & 0 & \sin(2 \cdot 0^2) \end{vmatrix} \]

This simplifies to:

\[ f(0) = \begin{vmatrix} \cos(0) & \sin(0) & -\cos(0) \\ \sin(0) & \cos(0) & \sin(0) \\ \sin(0) & 0 & \sin(0) \end{vmatrix} \]

Calculating the values:

\[ f(0) = \begin{vmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{vmatrix} \]

The determinant of a matrix with a row of zeros is 0. Thus,

\[ f(0) = 0 \]

### Step 2: Differentiate \( f(x) \)

To find \( f'(x) \), we will differentiate the determinant with respect to \( x \). We can use the Leibniz rule for differentiation of determinants:

\[ f'(x) = \frac{d}{dx} \begin{vmatrix} a_{11}(x) & a_{12}(x) & a_{13}(x) \\ a_{21}(x) & a_{22}(x) & a_{23}(x) \\ a_{31}(x) & a_{32}(x) & a_{33}(x) \end{vmatrix} \]

Using the formula for the derivative of a determinant:

\[ f'(x) = \begin{vmatrix} \frac{da_{11}}{dx} & a_{12} & a_{13} \\ \frac{da_{21}}{dx} & a_{22} & a_{23} \\ \frac{da_{31}}{dx} & a_{32} & a_{33} \end{vmatrix} + \begin{vmatrix} a_{11} & \frac{da_{12}}{dx} & a_{13} \\ a_{21} & \frac{da_{22}}{dx} & a_{23} \\ a_{31} & \frac{da_{32}}{dx} & a_{33} \end{vmatrix} + \begin{vmatrix} a_{11} & a_{12} & \frac{da_{13}}{dx} \\ a_{21} & a_{22} & \frac{da_{23}}{dx} \\ a_{31} & a_{32} & \frac{da_{33}}{dx} \end{vmatrix} \]

### Step 3: Evaluate \( f'(0) \)

Now we need to evaluate \( f'(0) \). We will find the derivatives of each element in the determinant at \( x = 0 \):

1. \( \frac{da_{11}}{dx} = -\sin(0) - 2 \cdot 0 \cdot \cos(0) = 0 \)
2. \( \frac{da_{12}}{dx} = \cos(0) + 2 \cdot 0 \cdot -\sin(0) = 1 \)
3. \( \frac{da_{13}}{dx} = \sin(0) + 2 \cdot 0 \cdot \cos(0) = 0 \)
4. \( \frac{da_{21}}{dx} = \cos(0) - 2 \cdot 0 \cdot \sin(0) = 1 \)
5. \( \frac{da_{22}}{dx} = -\sin(0) + 2 \cdot 0 \cdot \cos(0) = 0 \)
6. \( \frac{da_{23}}{dx} = \cos(0) - 2 \cdot 0 \cdot \sin(0) = 1 \)
7. \( \frac{da_{31}}{dx} = 2 \cos(0) = 2 \)
8. \( \frac{da_{32}}{dx} = 0 \)
9. \( \frac{da_{33}}{dx} = 2 \cdot 0 \cdot \cos(0) = 0 \)

Now substituting these into the determinant formula, we will find \( f'(0) \):

\[ f'(0) = \begin{vmatrix} 0 & 0 & 0 \\ 1 & 1 & 1 \\ 2 & 0 & 0 \end{vmatrix} + \begin{vmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{vmatrix} + \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{vmatrix} \]

Calculating these determinants:

1. The first determinant is \( 0 \).
2. The second determinant is also \( 0 \).
3. The third determinant is \( 1 \).

Thus,

\[ f'(0) = 0 + 0 + 1 = 1 \]

### Final Result

The value of \( f'(0) \) is:

\[ \boxed{1} \]