---
title: "Euler's theorem and tan⁻¹ function"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:euler-s-theorem-tan-inverse"
source: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-tan-inverse
site: BCT Engineering
---

# Euler's theorem and tan⁻¹ function

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2075 Baisakh, 1+4 marks
- Video: [Partial Differentiation - Euler's Theorem for Homogeneous Function](https://www.youtube.com/watch?v=bTs7ncA_AtY) — Dr.Gajendra Purohit and 2 more
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-tan-inverse
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

State Euler's Theorem for homogeneous function of two variables. If u = tan⁻¹((x³+y³)/(x+y)) then show that x ∂u/∂x + y ∂u/∂y = sin 2u.

## Short answer

Euler's theorem: if f is homogeneous of degree n, x ∂f/∂x + y ∂f/∂y = n f. Let z = tan u = (x³+y³)/(x+y); z is homogeneous of degree 2. Then x ∂z/∂x + y ∂z/∂y = 2z simplifies to x ∂u/∂x + y ∂u/∂y = sin 2u.

## Answer

#### Euler's theorem (1 mark)
If f(x, y) is homogeneous of degree *n*, then x ∂f/∂x + y ∂f/∂y = n f.

#### Proof (4 marks)
Given u = tan⁻¹((x³ + y³)/(x + y)), so

z = tan u = (x³ + y³)/(x + y).

Replacing x, y by tx, ty gives z(tx, ty) = t³(x³ + y³)/(t(x + y)) = t² z. So z is homogeneous of degree n = 2.

By Euler's theorem:

```text
x ∂z/∂x + y ∂z/∂y = 2z
```

As z = tan u: ∂z/∂x = sec²u ∂u/∂x, ∂z/∂y = sec²u ∂u/∂y. So

```text
sec²u ( x ∂u/∂x + y ∂u/∂y ) = 2 tan u
x ∂u/∂x + y ∂u/∂y = 2 tan u / sec²u
                   = 2 tan u cos²u
                   = 2 sin u cos u
                   = sin 2u
```

Hence proved. ∎

## Check yourself

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

1. For u = tan⁻¹((x³+y³)/(x+y)), which homogeneous function is used?
   A. z = tan⁻¹ u
   B. z = u
   C. z = tan u = (x³+y³)/(x+y)
   D. z = sin u
   Answer: C. u itself is not homogeneous; z = tan u is. (Key point: Function to use)

2. What is the degree of z = (x³+y³)/(x+y)?
   A. 2
   B. 1
   C. 3
   D. 4
   Answer: A. z(tx, ty) = t³/t · z = t² z. (Key point: Degree)

3. With Euler: x z_x + y z_y = 2z and z_x = sec²u u_x. What is x u_x + y u_y?
   A. 2 tan u / sec²u = sin 2u
   B. 2 sec²u
   C. 2 tan u sec²u
   D. tan u
   Answer: A. sec²u(x u_x + y u_y) = 2 tan u, so the sum = 2 tan u cos²u = 2 sin u cos u = sin 2u. (Key point: Chain rule)

## Related in Mathematics

- [Euler's theorem verified for xⁿ tan⁻¹(y/x)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aeuler-s-theorem-verified-x) — Mathematics, chapter 1
- [Euler's theorem statement with sin u example](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aeuler-s-theorem-statement-sin) — Mathematics, chapter 1, asked 3×
- [Euler's theorem statement with cos⁻¹ example](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aeuler-s-theorem-statement-cos) — Mathematics, chapter 1
- [Euler's theorem verified for (x^¼+y^¼)/(x^½+y^⅕)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aeuler-s-theorem-verified-x-2) — Mathematics, chapter 1
- [Homogeneous function: sin u=(√x−√y)/(√x+√y)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ahomogeneous-function-sin-u-sqrt) — Mathematics, chapter 1
- [Euler's theorem for cosec⁻¹ function](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aeuler-s-theorem-cosec-inverse) — Mathematics, chapter 1

## Related in other subjects

- [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 = e^(a tan⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aleibnitz-exp-atan) — Mathematics I, chapter 1, asked 2×
- [Limit of (sin x)^(tan x) as x→π](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Alimit-sinx-power-tanx-at-pi) — Mathematics I, chapter 1
- [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×