---
title: "Maclaurin expansion of e^(sin x)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:maclaurin-e-sin-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/maclaurin-e-sin-x
site: BCT Engineering
---

# Maclaurin expansion of e^(sin x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2067 Asoj
- Video: [Power Series Expansion of e^Sinx || Maclaurin's Series || Dr Prashant Patil](https://www.youtube.com/watch?v=xh2Au1kdJIE) — Dr Prashant Patil
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/maclaurin-e-sin-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Obtain the series expansion of e^(sin x) by Maclaurin's theorem as far as the term x⁴.

## Short answer

Compute derivatives at 0: f(0)=1, f'(0)=1, f''(0)=1, f'''(0)=0, f''''(0)=-3. Then e^(sin x) = 1 + x + x^2/2 - x^4/8 + ...

## Answer

Let f(x) = e^(sin x). Maclaurin's series: f(x) = f(0) + x f′(0) + (x²/2!) f″(0) + (x³/3!) f‴(0) + (x⁴/4!) f⁗(0) + ...

```text
f   = e^(sin x)
f′  = cos x · f
f″  = −sin x · f + cos²x · f            = f (cos²x − sin x)
f‴  = f′(cos²x − sin x) + f(−2 sin x cos x − cos x)
f⁗  = f″(cos²x − sin x) + 2f′(−2 sin x cos x − cos x)
        + f(−2cos 2x + sin x)
```
At x = 0 (sin 0 = 0, cos 0 = 1, f = 1):

```text
f(0)  = 1
f′(0) = 1
f″(0) = 1·(1 − 0) = 1
f‴(0) = 1·1 + 1·(0 − 1) = 0
f⁗(0) = 1·1 + 2·1·(−1) + 1·(−2) = −3
```
So

f(x) = 1 + x + x²/2 + 0·x³ − 3x⁴/24 + ...

**Answer: e^(sin x) = 1 + x + x²/2 − x⁴/8 + ...**

## Check yourself

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

1. For f = e^(sin x), f' =
   A. -cos x * f
   B. cos x * f
   C. f
   D. sin x * f
   Answer: B. Chain rule. (Key point: First derivative)

2. f(0), f'(0), f''(0) are:
   A. 0, 1, 1
   B. 1, 1, 1
   C. 1, 1, 0
   D. 1, 0, 1
   Answer: B. f(0) = 1, f' = 1, f'' = cos^2 - sin = 1. (Key point: Values)

3. f'''(0) =
   A. 0
   B. 2
   C. -1
   D. 1
   Answer: A. 1 + (0 - 1) = 0. (Key point: Third)

4. The series is:
   A. 1 + x + x^2 - x^4/8
   B. 1 + x - x^4/8
   C. 1 + x + x^2/2 + x^3/6
   D. 1 + x + x^2/2 - x^4/8
   Answer: D. -3/24 = -1/8. (Key point: Series)

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