Find the derivatives of each of the following functions :
`f(x)=8x^(4)`

Options

A. `30x^(2)`
B. `32x^(3)`
C. `32 x`
D. इनमे से कोई नहीं

Answer

To find the derivative of the function \( f(x) = 8x^4 \), we will use the power rule of differentiation. The power rule states that if \( y = x^n \), then the derivative \( \frac{dy}{dx} = n \cdot x^{n-1} \).

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = 8x^4 \]

2. **Apply the power rule**:
According to the power rule, the derivative of \( x^n \) is \( n \cdot x^{n-1} \). Here, \( n = 4 \).
\[ \frac{d}{dx}(x^4) = 4x^{4-1} = 4x^3 \]

3. **Multiply by the constant**:
Since there is a constant (8) multiplied by \( x^4 \), we multiply the derivative by this constant:
\[ f'(x) = 8 \cdot \frac{d}{dx}(x^4) = 8 \cdot 4x^3 \]

4. **Simplify the expression**:
\[ f'(x) = 32x^3 \]

### Final Answer:
The derivative of the function \( f(x) = 8x^4 \) is:
\[ f'(x) = 32x^3 \]