Find the derivatives of the following functions (1-3) at any point of their domains :
`y=ax^(3)+bx^(2)+cx+d`

Answer

To find the derivative of the function \( y = ax^3 + bx^2 + cx + d \), we will apply the power rule of differentiation. The power rule states that if \( y = x^n \), then the derivative \( \frac{dy}{dx} = nx^{n-1} \).

Let's go through the steps to find the derivative:

### Step 1: Differentiate each term of the function

1. **Differentiate \( ax^3 \)**:
- Using the power rule:
\[ \frac{d}{dx}(ax^3) = 3ax^{3-1} = 3ax^2 \]

2. **Differentiate \( bx^2 \)**:
- Again using the power rule:
\[ \frac{d}{dx}(bx^2) = 2bx^{2-1} = 2bx \]

3. **Differentiate \( cx \)**:
- The derivative of \( cx \) is:
\[ \frac{d}{dx}(cx) = c \]

4. **Differentiate \( d \)**:
- Since \( d \) is a constant, its derivative is:
\[ \frac{d}{dx}(d) = 0 \]

### Step 2: Combine the derivatives

Now, we combine all the derivatives we calculated:

\[ \frac{dy}{dx} = 3ax^2 + 2bx + c + 0 \]

Thus, the derivative of the function \( y = ax^3 + bx^2 + cx + d \) is:

\[ \frac{dy}{dx} = 3ax^2 + 2bx + c \]

### Final Answer:
\[ \frac{dy}{dx} = 3ax^2 + 2bx + c \]

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