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

# Derivatives of y = sin(m sin⁻¹x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2067 Asoj
- Video: [Bsc part 1 Successive Differentiation.If y=sin(msin−1x),Prove That (1−x2)y2−xy1+m2y=0](https://www.youtube.com/watch?v=QCgu6FddACY) — edu with sushant 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/derivatives-sin-m-asin-two-parts
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

If y = sin(m sin⁻¹x), prove that a) (1−x²)y₂ − xy₁ + m²y = 0, b) (1−x²)y_{n+2} − (2n+1)xy_{n+1} + (m²−n²)y_n = 0.

## Short answer

From y = sin(m sin⁻¹x), y₁ = m cos(m sin⁻¹x)/√(1−x²). Squaring gives (1−x²)y₁² = m²(1−y²); differentiating yields (1−x²)y₂ − xy₁ + m²y = 0. Applying Leibniz n times to this equation gives (1−x²)y_{n+2} − (2n+1)xy_{n+1} + (m²−n²)y_n = 0.

## Answer

Let y = sin(m sin⁻¹x). Then y₁ = m cos(m sin⁻¹x) / √(1 − x²).

#### (a)
(1 − x²) y₁² = m² cos²(m sin⁻¹x) = m²(1 − y²).

Differentiate: 2(1 − x²) y₁y₂ − 2x y₁² = −2m² y y₁. Divide by 2y₁:

**(1 − x²) y₂ − x y₁ + m² y = 0.** Proved.

#### (b)
Differentiate the result of (a) n times by Leibnitz's theorem:

```text
(1−x²)yₙ₊₂ + n(−2x)yₙ₊₁ + [n(n−1)/2](−2)yₙ
      − [x yₙ₊₁ + n yₙ] + m² yₙ = 0
```
Collect terms: the coefficient of yₙ is −n(n−1) − n + m² = m² − n².

**(1 − x²) yₙ₊₂ − (2n+1) x yₙ₊₁ + (m² − n²) yₙ = 0.** Proved.

## Check yourself

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

1. For y = sin(m sin^-1 x), y_1 =
   A. -m sin(...)
   B. m cos(m sin^-1 x)
   C. m cos(m sin^-1 x)/sqrt(1-x^2)
   D. cos(...)/sqrt(1-x^2)
   Answer: C. Chain rule gives the derivative of sin^-1 x as 1/sqrt(1-x^2). (Key point: First derivative)

2. Part (a) result:
   A. (1-x^2) y_2 + x y_1 + m^2 y = 0
   B. (1-x^2) y_2 - x y_1 + m^2 y = 0
   C. (1-x^2) y_2 - 2x y_1 = 0
   D. (1-x^2) y_2 - x y_1 - m^2 y = 0
   Answer: B. From (1-x^2) y_1^2 = m^2(1-y^2). (Key point: Part (a))

3. The y_n coefficient in (b) is:
   A. m^2 + n^2
   B. n^2 - m^2
   C. m^2 - n
   D. m^2 - n^2
   Answer: D. -n(n-1) - n + m^2. (Key point: Part (b))

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