Find `(a)dy/dxand(b)(d^(2)y)/(dx^(2))` when y is given by :
`ax^(3)+bx^(2)+cx+d`

Answer

To solve the problem, we need to find the first derivative \( \frac{dy}{dx} \) and the second derivative \( \frac{d^2y}{dx^2} \) of the function \( y = ax^3 + bx^2 + cx + d \).

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Given:
\[ y = ax^3 + bx^2 + cx + d \]

To find \( \frac{dy}{dx} \), we differentiate \( y \) with respect to \( x \):

1. The derivative of \( ax^3 \) is \( 3ax^2 \) (using the power rule).
2. The derivative of \( bx^2 \) is \( 2bx \).
3. The derivative of \( cx \) is \( c \).
4. The derivative of the constant \( d \) is \( 0 \).

Combining these results, we have:
\[ \frac{dy}{dx} = 3ax^2 + 2bx + c \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now, we differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \):

1. The derivative of \( 3ax^2 \) is \( 6ax \).
2. The derivative of \( 2bx \) is \( 2b \).
3. The derivative of \( c \) is \( 0 \).

Combining these results, we have:
\[ \frac{d^2y}{dx^2} = 6ax + 2b \]

### Final Answers:
(a) The first derivative \( \frac{dy}{dx} = 3ax^2 + 2bx + c \)

(b) The second derivative \( \frac{d^2y}{dx^2} = 6ax + 2b \)

---