To differentiate the function \( y = ax + bx^2 + c \) with respect to \( x \), we will use the power rule of differentiation.

Given:
\[
y = ax + bx^2 + c
\]
where:
- \( a \), \( b \), and \( c \) are constants.

### Steps for differentiation:
1. The derivative of \( ax \) with respect to \( x \) is \( a \).
2. The derivative of \( bx^2 \) with respect to \( x \) is \( 2bx \) (using the power rule).
3. The derivative of the constant \( c \) is \( 0 \), as the derivative of any constant is zero.

Thus, the derivative \( \frac{dy}{dx} \) is:

\[
\frac{dy}{dx} = a + 2bx
\]

### Final answer:
\[
\frac{dy}{dx} = a + 2bx
\]

Would you like any further details or have any questions?

### 5 Related Questions:
1. How do you apply the power rule for differentiation?
2. What is the derivative of higher degree polynomials like \( bx^n \)?
3. How does the chain rule apply when differentiating composite functions?
4. What is the physical significance of the derivative in terms of rate of change?
5. How can we find the second derivative of the function \( y = ax + bx^2 + c \)?

### Tip:
When differentiating polynomials, apply the power rule to each term individually and sum up the results for a smooth process.

Understand the step-by-step differentiation of the polynomial function y = ax + bx^2 + c using the power rule. This detailed solution breaks down each term's derivative, explaining how to handle constants and polynomial terms. Perfect for students in Grades 9-12.

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