Find `(dy)/(dx)` if `y=8x^6-4x^3`.

Options

A. `24x^5-8x^2`
B. `48x^5-12x^2`
C. `4x^5-x^2`
D. `3x^5 -2x^2`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = 8x^6 - 4x^3\), we will use the power rule of differentiation. The power rule states that if \(y = x^n\), then \(\frac{dy}{dx} = n \cdot x^{n-1}\).

### Step-by-step solution:

1. **Identify the function**:
\[ y = 8x^6 - 4x^3 \]

2. **Differentiate each term separately**:
- For the first term \(8x^6\):
- Using the power rule:
\[ \frac{d}{dx}(8x^6) = 8 \cdot 6x^{6-1} = 48x^5 \]
- For the second term \(-4x^3\):
- Again using the power rule:
\[ \frac{d}{dx}(-4x^3) = -4 \cdot 3x^{3-1} = -12x^2 \]

3. **Combine the derivatives**:
\[ \frac{dy}{dx} = 48x^5 - 12x^2 \]

### Final Answer:
\[ \frac{dy}{dx} = 48x^5 - 12x^2 \]