Q.Find the second order derivative of the function: $x^{20}$

Differentiate $x^{20}$ twice using the power rule $\frac{d}{dx}x^n = n\,x^{n-1}$ — this is successive (repeated) differentiation.

First derivative:

Second derivative — differentiate the result again:

A second-order derivative just means we differentiate, then differentiate the result again. For a pure power of $x$ the only tool we need is the power rule, $\frac{d}{dx}x^n = n\,x^{n-1}$ — no chain rule, no product rule.

Step 1 — First derivative

For $y = x^{20}$,

Step 2 — Second derivative

Now differentiate $20x^{19}$. The constant $20$ stays put; apply the power rule to $x^{19}$:

Step 3 — Simplify

Since $20\times 19 = 380$,

Method: Successive Differentiation of a Power Function

This method finds a second (or higher) order derivative of a pure power $x^n$ by applying the power rule repeatedly, one order at a time.

Steps

Step 1: Identify the function type and the order of derivative required

Check whether the expression is a pure power of $x$ (possibly with a constant coefficient). For a pure power, no chain rule, product rule, or quotient rule is needed — only the power rule, applied as many times as the required order.

Step 2: Differentiate once to get the first derivative

Apply the power rule to the original function to obtain $y_1 = \dfrac{dy}{dx}$. This reduces the exponent by 1 and multiplies by the original exponent.

Step 3: Differentiate the result again for the second derivative

Treat $y_1$ as a new function and apply the power rule to it directly — differentiate the coefficient-power expression, not the original function. This gives $y_2 = \dfrac{d^2y}{dx^2}$.

Step 4: Simplify the constant multiplier

Multiply out any numerical coefficients that arise from repeated application (e.g., $n(n-1)$) and leave the answer as a single coefficient times a power of $x$. For higher orders, keep repeating Steps 2–3, tracking how the exponent decreases and the coefficient grows by successive multiplication.

Common Mistakes

Mistake 1: Stopping after computing only the first derivative

Why it's wrong: the question asks for the second order derivative, but $20x^{19}$ is only an intermediate step. Correct approach: always re-read what order is asked, and explicitly differentiate the first-derivative expression once more before writing the final answer.

Mistake 2: Forgetting to multiply the existing coefficient into the new one

Why it's wrong: when differentiating $20x^{19}$, both the coefficient $20$ and the exponent $19$ must be multiplied together (giving $380$) — some students only bring down the exponent and forget to multiply it by the coefficient already present. Correct approach: apply the power rule to the whole term as $20\cdot 19\,x^{18}$, not just $x^{18}$.

Mistake 3: Confusing $\dfrac{d^2y}{dx^2}$ with $\left(\dfrac{dy}{dx}\right)^2$

Why it's wrong: squaring the first derivative gives a completely different (and wrong) expression — the second derivative is the derivative of the derivative, not its square. Correct approach: always compute the second derivative by differentiating the first-derivative expression again, never by squaring it.