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**:
We have \(y = 3x^3 - 4x^2\).

2. **Differentiate each term**:
We will differentiate \(3x^3\) and \(-4x^2\) separately using the power rule, which states that \(\frac{d}{dx}(x^n) = n \cdot x^{n-1}\).

- For the first term \(3x^3\):
\[ \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2 \]

- For the second term \(-4x^2\):
\[ \frac{d}{dx}(-4x^2) = -4 \cdot 2x^{2-1} = -8x \]

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

4. **Final result**:
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 9x^2 - 8x \]