---
title: "Total derivative of sin(x/y)"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:total-derivative-sin-x-y"
source: https://www.bctengineer.com/q/semester_2/mathematics/total-derivative-sin-x-y
site: BCT Engineering
---

# Total derivative of sin(x/y)

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2070 Magh, 5 marks
- Video: [How to find total derivative |𝒖=𝑺𝒊𝒏(𝒚/𝒙) 𝒙=𝒆^𝒕  𝒚=𝒕^𝟐 || Partial Differentiation|| Dr Prashant Patil](https://www.youtube.com/watch?v=B3YbaEFNzu0) — Dr Prashant Patil
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/total-derivative-sin-x-y
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

Find du/dt if u = sin(x/y), x = eᵗ and y = t².

## Short answer

Using chain rule, du/dt = (1/y)cos(x/y)·e^t − (x/y²)cos(x/y)·2t. With x=e^t, y=t², du/dt = e^t (t−2) cos(e^t/t²) / t³.

## Answer

Here u = sin(x/y) is a function of x and y, which are themselves functions of t. By the chain rule for total derivatives:

du/dt = (∂u/∂x)(dx/dt) + (∂u/∂y)(dy/dt)

#### Partial derivatives
```text
∂u/∂x = cos(x/y) · (1/y)      = cos(x/y)/y
∂u/∂y = cos(x/y) · (−x/y²)    = −x cos(x/y)/y²
```

#### Derivatives of x and y
x = eᵗ → dx/dt = eᵗ;  y = t² → dy/dt = 2t.

#### Substitute
```text
du/dt = cos(x/y)/y · eᵗ  +  ( −x cos(x/y)/y² ) · 2t
      = cos(eᵗ/t²) [ eᵗ/t²  −  eᵗ · 2t / t⁴ ]
      = cos(eᵗ/t²) [ eᵗ/t²  −  2eᵗ/t³ ]
      = eᵗ (t − 2)/t³ · cos(eᵗ/t²)
```

**Answer: du/dt = (t − 2) eᵗ cos(eᵗ/t²) / t³.**

## Check yourself

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

1. The total derivative formula for u(x, y) with x, y functions of t is:
   A. du/dt = u_x + u_y
   B. du/dt = u_x · u_y
   C. du/dt = (dx/dt)(dy/dt)
   D. du/dt = (∂u/∂x)(dx/dt) + (∂u/∂y)(dy/dt)
   Answer: D. Each partial derivative is multiplied by the rate of change of its variable. (Key point: Chain rule)

2. ∂u/∂y for u = sin(x/y) is:
   A. −x cos(x/y)/y²
   B. −cos(x/y)/y²
   C. x cos(x/y)/y²
   D. cos(x/y)/y
   Answer: A. d/dy(x/y) = -x/y², times cos(x/y). (Key point: Partial derivative)

3. The final result for du/dt is:
   A. (t+2) eᵗ cos(eᵗ/t²)/t³
   B. eᵗ cos(eᵗ/t²)/t²
   C. (t−2) eᵗ cos(eᵗ/t²)/t³
   D. (t−2) eᵗ sin(eᵗ/t²)/t³
   Answer: C. cos(eᵗ/t²)[eᵗ/t² − 2eᵗ/t³] = eᵗ(t−2)cos(eᵗ/t²)/t³. (Key point: Result)

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