Find `(dy)/(dx)` if `y =3x^3-4x^2`.

Options

A. `3/4 x^4 - 4/3 x^3`
B. `12x^2-8x`
C. `9x^2-8x`
D. `16x^2-8x`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = 3x^3 - 4x^2\), we will differentiate the function with respect to \(x\) using the power rule of differentiation.

### Step-by-step solution:

1. **Identify the function**:
\[ y = 3x^3 - 4x^2 \]

2. **Apply the power rule**:
The power rule states that if \(y = ax^n\), then \(\frac{dy}{dx} = n \cdot ax^{n-1}\). We will apply this rule to each term in the function.

3. **Differentiate the first term \(3x^3\)**:
- Here, \(a = 3\) and \(n = 3\).
- Applying the power rule:
\[ \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2 \]

4. **Differentiate the second term \(-4x^2\)**:
- Here, \(a = -4\) and \(n = 2\).
- Applying the power rule:
\[ \frac{d}{dx}(-4x^2) = -4 \cdot 2x^{2-1} = -8x \]

5. **Combine the results**:
Now, we combine the derivatives of both terms:
\[ \frac{dy}{dx} = 9x^2 - 8x \]

Thus, the final result is:
\[ \frac{dy}{dx} = 9x^2 - 8x \]