Q.Find the second order derivative of the function $x^2 + 3x + 2$.

The key idea is that since this is an explicit polynomial, we simply differentiate twice in the usual way — no implicit differentiation is needed here.

Step 1: Differentiate once.

$\frac{d}{dx}(x^2 + 3x + 2) = 2x + 3$

Step 2: Differentiate the result to get the second derivative.

$\frac{d}{dx}(2x + 3) = 2$

The core idea here is straightforward: differentiation is a linear operation — you can differentiate each term separately and add the results. For a polynomial like $x^2 + 3x + 2$, the power rule $\frac{d}{dx} x^n = n x^{n-1}$ is all you need. The second derivative is just the derivative of the first derivative, so we apply the rule twice.

Let’s walk through it.

1. First derivative. Differentiate each term:
   • $\frac{d}{dx}(x^2) = 2x^{2-1} = 2x$
   • $\frac{d}{dx}(3x) = 3 \cdot \frac{d}{dx}(x) = 3 \cdot 1 = 3$
   • $\frac{d}{dx}(2) = 0$ (derivative of a constant is zero) So the first derivative is:

2. Second derivative. Now differentiate $f'(x) = 2x + 3$:
   • $\frac{d}{dx}(2x) = 2 \cdot 1 = 2$
   • $\frac{d}{dx}(3) = 0$ Hence:

Method: Finding a Second-Order Derivative

The second derivative $\dfrac{d^2y}{dx^2}$ (written $y_2$) is simply the derivative of the first derivative — differentiate once, then differentiate the result again.

Steps

Step 1: Find the first derivative $y_1=\dfrac{dy}{dx}$

Differentiate the given function term by term using the power rule.

Step 2: Differentiate $y_1$ again to get $y_2$

Treat $y_1$ as a new function and apply the same term-by-term differentiation.

Step 3 (Applying to this problem): for the polynomial $x^2+3x+2$, the first derivative is $2x+3$; differentiating again, the $2x$ term gives $2$ and the constant $3$ vanishes, leaving $y_2=2$. For any quadratic $ax^2+bx+c$, this generalises to $y_2=2a$ always.

Common Mistakes

Mistake 1: Forgetting that the derivative of a constant is zero.

Why it's wrong: on the first pass the constant $2$ in $x^2+3x+2$ must vanish, and on the second pass the constant $3$ (left over from $2x+3$) must vanish too — carrying either one forward gives a wrong non-constant "second derivative". Correct approach: explicitly write $\frac{d}{dx}(\text{constant})=0$ at each stage rather than skipping it mentally.

Mistake 2: Stopping after only one differentiation.

Why it's wrong: the question asks for the second order derivative, but a rushed reading sometimes reports $2x+3$ (the first derivative) as the final answer. Correct approach: always count differentiation passes against the order asked for before writing the boxed answer.