Q.Find the second order derivative of the function $\tan^{-1} x$.

Concept: Successive Differentiation — the second-order derivative is found by differentiating the first derivative again.

Step 1: $y = \tan^{-1}x \Rightarrow y_1 = \dfrac{1}{1+x^2} = (1+x^2)^{-1}$. …

Step 1 — First derivative

Using the standard formula $\dfrac{d}{dx}\tan^{-1}x=\dfrac{1}{1+x^2}$:

Step 2 — Differentiate again (second order)

Write $y_1=(1+x^2)^{-1}$ and apply the chain rule together with the power rule:

$y_2=\frac{d}{dx}\left[(1+x^2)^{-1}\right]=-1\cdot(1+x^2)^{-2}\cdot\frac{d}{dx}(1+x^2)=-(1+x^2)^{-2}\cdot2x.$ …

Method: Finding a Second-Order Derivative of an Inverse Trig Function

The second derivative of $\tan^{-1}x$ is found the same way as any other second-order derivative: differentiate once using the standard inverse-trig derivative, then differentiate that result again using the chain rule on a negative power.

Steps

Step 1: Find the first derivative

Step 2: Differentiate $y_1$ again using the chain rule

Treat $(1+x^2)^{-1}$ as a composite function: the outer function is $u^{-1}$ (derivative $-u^{-2}$) and the inner function is $u=1+x^2$ (derivative $2x$).

$y_2 = -(1+x^2)^{-2}\cdot 2x = -\frac{2x}{(1+x^2)^2}.$ …

Common Mistakes

Mistake 1: Forgetting the chain rule when differentiating $(1+x^2)^{-1}$ a second time.

Why it's wrong: writing $y_2=-(1+x^2)^{-2}$ without the extra factor $2x$ (the derivative of the inner $1+x^2$) misses half the required chain-rule expansion. Correct approach: always differentiate the inner function $1+x^2$ explicitly and multiply it in.

Mistake 2: Confusing "second derivative" with "principal value branch". …