Q.If $y = \sin^{-1} x$, show that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 0$.

Concept: Leibniz Notation Derivatives — we differentiate $\sin^{-1}x$ twice and substitute into the given expression.

Step 1: First derivative.

If $y = \sin^{-1} x$, then

Step 2: Second derivative.

Differentiate again using the chain rule:

Step 3: Substitute into the left-hand side. …

The key insight here is that when you have an inverse trigonometric function, its derivative is a rational expression involving a square root. Differentiating that derivative gives a second derivative that, when combined with the first derivative, cancels out neatly. This is a classic verification problem — it tests your comfort with chain rule and algebraic manipulation in Leibniz notation.

Let’s work through it step by step.

1. First derivative Given $y = \sin^{-1} x$, we know

This is valid for $|x| < 1$. The domain matters because the square root must be real.

2. Second derivative Differentiate $\frac{dy}{dx}$ with respect to $x$. Write it as $(1 - x^2)^{-1/2}$ for easier differentiation:

The $-2x$ comes from the derivative of $(1 - x^2)$ by the chain rule. Simplify:

3. Form the expression We need to compute $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx}$. Substitute the derivatives:

4. Simplify term by term

   First term: $(1 - x^2) \cdot \frac{x}{(1 - x^2)^{3/2}} = \frac{x}{(1 - x^2)^{1/2}} = \frac{x}{\sqrt{1 - x^2}}$

   Second term: $x \cdot \frac{1}{\sqrt{1 - x^2}} = \frac{x}{\sqrt{1 - x^2}}$

5. Subtract

   $\frac{x}{\sqrt{1 - x^2}} - \frac{x}{\sqrt{1 - x^2}} = 0$ …

Method: Differentiating an Inverse Trig Function Twice and Verifying a Relation

This method applies whenever $y$ is an inverse trigonometric function (like $\sin^{-1}x$) and you must show a given combination of $y$, $\dfrac{dy}{dx}$, and $\dfrac{d^2y}{dx^2}$ equals a specific value (often $0$).

Steps

Step 1: Write down the known first-derivative formula

For $y=\sin^{-1}x$:

Recasting the square root as a negative fractional power makes the next differentiation cleaner.

Step 2: Differentiate again using the chain rule

Differentiate $(1-x^2)^{-1/2}$ by bringing down the power, reducing the exponent by $1$, and multiplying by the derivative of the inner function $(1-x^2)$, which is $-2x$: …

Common Mistakes

Mistake 1: Losing track of the sign when differentiating $(1-x^2)^{-1/2}$

Why it's wrong: the power rule contributes a negative sign (from the $-1/2$ exponent) and the chain rule contributes another negative sign (from the derivative of $1-x^2$ being $-2x$) — the two negatives multiply to a positive, but it is common to keep only one of them and end up with the wrong overall sign on $\dfrac{d^2y}{dx^2}$. Correct approach: write both negative signs out explicitly, side by side, before multiplying them.

Mistake 2: Exponent arithmetic error going from $-\tfrac12-1$ to $-\tfrac32$ …