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 \( y = ax^n \), then \( \frac{dy}{dx} = nax^{n-1} \).

### Step-by-Step Solution:

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

2. **Differentiate the first term \( 8x^6 \)**:
- Apply the power rule:
\[ \frac{d}{dx}(8x^6) = 6 \cdot 8x^{6-1} = 48x^5 \]

3. **Differentiate the second term \( -4x^3 \)**:
- Again, apply the power rule:
\[ \frac{d}{dx}(-4x^3) = 3 \cdot (-4)x^{3-1} = -12x^2 \]

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

5. **Final answer**:
\[ \frac{dy}{dx} = 48x^5 - 12x^2 \]