Find the first derivative & second derivative of given functions w.r.t. corresponding independent variable.
`y=6x^(2)-10x`

Options

A. `12x-10, 12`
B. `12x-x, 12`
C. `12x^(3)-10x^(2),12x`
D. 0, 0

Answer

To find the first and second derivatives of the function \( y = 6x^2 - 10x \) with respect to \( x \), we will follow these steps:

### Step 1: Find the First Derivative
The first derivative of a function \( y \) with respect to \( x \) is denoted as \( \frac{dy}{dx} \).

1. Start with the function:
\[ y = 6x^2 - 10x \]

2. Differentiate each term:
- For the term \( 6x^2 \):
- Use the power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \).
- Here, \( n = 2 \), so:
\[ \frac{d}{dx}(6x^2) = 6 \cdot 2x^{2-1} = 12x \]
- For the term \( -10x \):
- Again, use the power rule where \( n = 1 \):
\[ \frac{d}{dx}(-10x) = -10 \cdot 1x^{1-1} = -10 \]

3. Combine the results:
\[ \frac{dy}{dx} = 12x - 10 \]

### Step 2: Find the Second Derivative
The second derivative is denoted as \( \frac{d^2y}{dx^2} \).

1. Start with the first derivative:
\[ \frac{dy}{dx} = 12x - 10 \]

2. Differentiate the first derivative:
- For the term \( 12x \):
\[ \frac{d}{dx}(12x) = 12 \]
- For the constant term \( -10 \):
\[ \frac{d}{dx}(-10) = 0 \]

3. Combine the results:
\[ \frac{d^2y}{dx^2} = 12 \]

### Final Results
- The first derivative \( \frac{dy}{dx} = 12x - 10 \)
- The second derivative \( \frac{d^2y}{dx^2} = 12 \)