if y= sin mx, them the value of the determinant
`|{:(y,,y_(1),,y_(2)),(y_(3),,y_(4),,y_(5)),(y_(6),,y_(7),,y_(8)):}|" Where " y_(n)=(d^(n)y)/(dx^(n)) " is "`

Options

A. `m^(9)`
B. `m^(2)`
C. `m^(3)`
D. 0

Explanation

To solve the given determinant problem step by step, we will follow the derivatives of \( y = \sin(mx) \) and then evaluate the determinant.

### Step 1: Calculate the derivatives of \( y \)
Given \( y = \sin(mx) \), we can find the first few derivatives:
- \( y_1 = \frac{dy}{dx} = m \cos(mx) \)
- \( y_2 = \frac{d^2y}{dx^2} = -m^2 \sin(mx) \)
- \( y_3 = \frac{d^3y}{dx^3} = -m^3 \cos(mx) \)
- \( y_4 = \frac{d^4y}{dx^4} = m^4 \sin(mx) \)
- \( y_5 = \frac{d^5y}{dx^5} = m^5 \cos(mx) \)
- \( y_6 = \frac{d^6y}{dx^6} = -m^6 \sin(mx) \)
- \( y_7 = \frac{d^7y}{dx^7} = -m^7 \cos(mx) \)
- \( y_8 = \frac{d^8y}{dx^8} = m^8 \sin(mx) \)

### Step 2: Form the determinant
Now we can form the determinant using these derivatives:
\[ D = \begin{vmatrix} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{vmatrix} = \begin{vmatrix} \sin(mx) & m \cos(mx) & -m^2 \sin(mx) \\ -m^3 \cos(mx) & m^4 \sin(mx) & m^5 \cos(mx) \\ -m^6 \sin(mx) & -m^7 \cos(mx) & m^8 \sin(mx) \end{vmatrix} \]

### Step 3: Factor out common terms
We can factor out common terms from each row:
- From the first row, we can factor out \( \sin(mx) \).
- From the second row, we can factor out \( m^4 \).
- From the third row, we can factor out \( -m^6 \).

Thus, we get:
\[ D = \sin(mx) \cdot m^4 \cdot (-m^6) \cdot \begin{vmatrix} 1 & \frac{m \cos(mx)}{\sin(mx)} & -m^2 \\ -m^3 \frac{\cos(mx)}{\sin(mx)} & m^4 & m^5 \frac{\cos(mx)}{\sin(mx)} \\ -m^6 & -m^7 \frac{\cos(mx)}{\sin(mx)} & m^8 \end{vmatrix} \]

### Step 4: Simplify the determinant
Now we will simplify the determinant:
\[ D = -m^{10} \sin(mx) \begin{vmatrix} 1 & m \cot(mx) & -m^2 \\ -m^3 \cot(mx) & m^4 & m^5 \cot(mx) \\ -m^6 & -m^7 \cot(mx) & m^8 \end{vmatrix} \]

### Step 5: Evaluate the determinant
Calculating the determinant will involve applying the properties of determinants. However, we can observe that the rows are linearly dependent because the functions \( \sin(mx) \) and \( \cos(mx) \) repeat in a periodic manner. This indicates that the determinant will equal zero.

### Final Answer
Thus, the value of the determinant is:
\[ D = 0 \]