---
title: "Second derivatives of r with x=r cosθ, y=r sinθ"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:second-derivatives-r-x-r"
source: https://www.bctengineer.com/q/semester_2/mathematics/second-derivatives-r-x-r
site: BCT Engineering
---

# Second derivatives of r with x=r cosθ, y=r sinθ

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2071 Magh, 5 marks
- Video: [If x = r cos theta , y = r sin theta , show that ∂r/∂x = ∂x/∂r](https://www.youtube.com/watch?v=nb2mSVwD03g) — Nitin Kumar Gupta Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/second-derivatives-r-x-r
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

If x = r cos θ, y = r sin θ. Prove that ∂²r/∂x² + ∂²r/∂y² = (1/r)[(∂r/∂x)² + (∂r/∂y)²]. [Right-hand side as printed; scan unclear]

## Short answer

From r²=x²+y², ∂r/∂x = x/r, ∂r/∂y = y/r. Second derivatives: ∂²r/∂x² = y²/r³, ∂²r/∂y² = x²/r³, sum = 1/r. The right side equals (1/r)((x²+y²)/r²) = 1/r, verifying the identity.

## Answer

**Reading of the question.** The scan was unclear about the right-hand side. I read it as printed: ∂²r/∂x² + ∂²r/∂y² = (1/r)[(∂r/∂x)² + (∂r/∂y)²], and this is a correct identity (both sides equal 1/r), so it is proved as printed.

Here x = r cos θ, y = r sin θ, so r² = x² + y², r = √(x² + y²).

#### First derivatives
Differentiating r² = x² + y²: 2r ∂r/∂x = 2x, so

```text
∂r/∂x = x/r = cos θ
∂r/∂y = y/r = sin θ
```

#### Second derivatives
```text
∂²r/∂x² = ∂/∂x (x/r) = 1/r − (x/r²)(∂r/∂x) = 1/r − x²/r³ = y²/r³
∂²r/∂y² = ∂/∂y (y/r) = 1/r − y²/r³ = x²/r³
```

Adding:

```text
∂²r/∂x² + ∂²r/∂y² = (x² + y²)/r³ = r²/r³ = 1/r        ... (1)
```

#### Right-hand side
```text
(∂r/∂x)² + (∂r/∂y)² = x²/r² + y²/r² = 1
(1/r)[(∂r/∂x)² + (∂r/∂y)²] = 1/r                       ... (2)
```

From (1) and (2):

∂²r/∂x² + ∂²r/∂y² = (1/r)[(∂r/∂x)² + (∂r/∂y)²]. ∎

## Check yourself

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

1. With r² = x²+y², ∂r/∂x equals:
   A. r/x
   B. x r
   C. 1/r
   D. x/r
   Answer: D. 2r r_x = 2x gives r_x = x/r = cos θ. (Key point: First derivatives)

2. What is ∂²r/∂x²?
   A. x²/r³
   B. y²/r³
   C. 1/r
   D. −y²/r³
   Answer: B. ∂/∂x(x/r) = 1/r − x²/r³ = y²/r³. (Key point: Second derivatives)

3. Both sides of the identity equal:
   A. r
   B. 1/r
   C. r²
   D. 1
   Answer: B. LHS = (x²+y²)/r³ = 1/r; RHS = (1/r)(cos²θ+sin²θ) = 1/r. (Key point: Equating sides)

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