---
title: "Leibnitz theorem: y = e^(a tan⁻¹x)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:leibnitz-exp-atan"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/leibnitz-exp-atan
site: BCT Engineering
---

# Leibnitz theorem: y = e^(a tan⁻¹x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Asoj; 2067 Asar (asked 2 times)
- Video: [If y=e^tan^-1(x),prove that  (1+x^2)Yn+2+(2(n+1)x-1)Yn+1 + n(n+1)Yn=0{based on Leibnitz Theorem}](https://www.youtube.com/watch?v=rtDu5QW0geY) — KSB Maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/leibnitz-exp-atan
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

State Leibnitz's theorem on higher order derivatives. If y = e^(a tan⁻¹x), prove that (1+x²)y_{n+2} + (2nx+2x−a)y_{n+1} + n(n+1)y_n = 0.

## Short answer

Leibnitz’s theorem: (uv)ₙ = Σ_{r=0}^n C(n,r) u_{n−r} v_r. For y = e^{a tan⁻¹x}, differentiate to get (1+x²)y₁ = a y. Applying the theorem (n+1) times yields (1+x²)y_{n+2} + (2nx+2x−a)y_{n+1} + n(n+1)y_n = 0.

## Answer

**Leibnitz's theorem.** If u and v are functions of x with nth derivatives, then the nth derivative of the product is

```text
(uv)ₙ = uₙv + C(n,1)uₙ₋₁v₁ + C(n,2)uₙ₋₂v₂ + ... + u vₙ
```
where C(n,r) = n!/(r!(n−r)!).

#### Proof
y = e^(a tan⁻¹x). Then

y₁ = e^(a tan⁻¹x) · a/(1+x²) = a y/(1+x²)

**(1 + x²) y₁ = a y** …(1)

Differentiate (1) (n+1) times by Leibnitz's theorem. Take u = y₁ (or y), v = 1 + x² (v₁ = 2x, v₂ = 2):

```text
(1+x²)yₙ₊₂ + (n+1)(2x)yₙ₊₁ + [(n+1)n/2](2)yₙ = a yₙ₊₁
```
That is

**(1 + x²) yₙ₊₂ + (2nx + 2x − a) yₙ₊₁ + n(n+1) yₙ = 0.** Proved.

## Check yourself

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

1. Leibnitz's theorem gives (uv)_n = u_n v + C(n,1)u_{n-1}v_1 + ... What does it compute?
   A. The nth integral of a sum
   B. The nth root of a product
   C. The nth derivative of a product
   D. The nth derivative of a quotient
   Answer: C. It is the product rule for the nth derivative with binomial coefficients. (Key point: Leibnitz statement)

2. For y = e^(a tan^-1 x), which relation holds?
   A. (1+x^2) y_1 = a y
   B. y_1 = a y
   C. (1-x^2) y_1 = a y
   D. (1+x^2) y_1 = y/a
   Answer: A. y_1 = a y/(1+x^2). (Key point: Base relation)

3. How many times is (1+x^2) y_1 = a y differentiated to reach y_{n+2}?
   A. n times
   B. n+1 times
   C. 2n times
   D. n+2 times
   Answer: B. y_1 differentiated n+1 times becomes y_{n+2}. (Key point: Number of differentiations)

4. The y_n coefficient from [(n+1)n/2](2) is:
   A. (n+1)/2
   B. n^2
   C. 2n
   D. n(n+1)
   Answer: D. [(n+1)n/2] * 2 = n(n+1). (Key point: Coefficient of y_n)

## Related in Mathematics I

- [Leibnitz theorem: log y = tan⁻¹x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aleibnitz-log-atan) — Mathematics I, chapter 1, asked 2×
- [Leibnitz theorem: y = sin(m sin⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aleibnitz-sin-m-asin) — Mathematics I, chapter 1, asked 2×
- [Leibnitz theorem: y = (x²−1)ⁿ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aleibnitz-x2-minus-1-power-n) — Mathematics I, chapter 1, asked 2×
- [Leibnitz theorem: y = e^(x²)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aleibnitz-exp-x-squared) — Mathematics I, chapter 1, asked 2×
- [Leibnitz theorem: y = (sin⁻¹x)²](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aleibnitz-asin-squared) — Mathematics I, chapter 1
- [Derivatives of y^(1/m) + y^(−1/m) = 2x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aderivatives-y-power-m-sum) — Mathematics I, chapter 1

## Related in other subjects

- [Euler's theorem and tan⁻¹ function](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aeuler-s-theorem-tan-inverse) — Mathematics, chapter 1
- [Techniques for higher order active filters](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ahigher-order-active-techniques) — Filter Design, chapter 8
- [Why decompose higher-order filters; ladder realization](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adecomposition-ladder-scheme) — Digital Control System, chapter 4
- [Residue theorem: ∮tan z dz on |z|=2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-tan-z) — Applied Mathematics, chapter 1