---
title: "Derivatives of y = (sin⁻¹x)²"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:leibnitz-asin-squared-two-parts"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/leibnitz-asin-squared-two-parts
site: BCT Engineering
---

# Derivatives of y = (sin⁻¹x)²

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2072 Kartik
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/leibnitz-asin-squared-two-parts
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

If y = (sin⁻¹x)², then show that: i) (1−x²)y₂ − xy₁ − 2 = 0, ii) (1−x²)y_{n+2} − (2n+1)xy_{n+1} − n²y_n = 0.

## Short answer

Differentiating (1-x²)y₁² = 4y gives (1-x²)y₂ - xy₁ - 2 = 0. Applying Leibnitz's theorem n times to this produces (1-x²)y_{n+2} - (2n+1)xy_{n+1} - n²y_n = 0.

## Answer

Let y = (sin⁻¹x)². Then y₁ = 2 sin⁻¹x / √(1 − x²), so (1 − x²) y₁² = 4y.

#### (i)
Differentiate (1 − x²) y₁² = 4y with respect to x:

2(1 − x²) y₁y₂ − 2x y₁² = 4 y₁

Divide by 2y₁:

**(1 − x²) y₂ − x y₁ − 2 = 0.** Proved.

#### (ii)
Differentiate the result of (i) n times using Leibnitz's theorem (the constant 2 gives 0):

```text
(1−x²)yₙ₊₂ + n(−2x)yₙ₊₁ + [n(n−1)/2](−2)yₙ − x yₙ₊₁ − n yₙ = 0
(1−x²)yₙ₊₂ − (2n+1)x yₙ₊₁ − n² yₙ = 0
```
since −n(n−1) − n = −n². **Proved.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Part (i) begins from:
   A. (1-x^2) y_1 = 4y
   B. y_1 = 4y
   C. (1-x^2) y_1^2 = 2y
   D. (1-x^2) y_1^2 = 4y
   Answer: D. y_1^2 = 4(sin^-1 x)^2/(1-x^2). (Key point: Starting relation)

2. Dividing by 2y_1 yields:
   A. (1-x^2) y_2 - x y_1 - 2 = 0
   B. y_2 = 2
   C. (1-x^2) y_2 - x y_1 = 0
   D. (1-x^2) y_2 + 2 = 0
   Answer: A. 4y_1/(2y_1) = 2. (Key point: Why divide)

3. Part (ii) uses:
   A. Maclaurin series
   B. Integration of (i)
   C. Rolle's theorem
   D. Leibnitz on the result of (i)
   Answer: D. Differentiate (i) n times. (Key point: Part (ii))

4. The y_n coefficient: -n(n-1) - n =
   A. -2n
   B. n^2
   C. -n^2
   D. -n^2 - n
   Answer: C. = -n^2 + n - n. (Key point: Coefficient)

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