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Name: Find dy/dx by implicit differentiation. xey=x−y
Uploaded: 2022-02-23T17:22:57+00:00
Description: Find dy/dx by implicit differentiation. xey=x−y

Chesley

Answered question

2021-10-04

Find dy/dx by implicit differentiation.

x e^{y} = x - y

Answer & Explanation

lobeflepnoumni

Skilled2021-10-05Added 99 answers

x e^{y} = x - y

Differentiate throughout with respect to x

\frac{d (x e^{y})}{d x} = \frac{d x}{d x} - \frac{d y}{d x}

\underset{PRODUCT RULE}{\underset{⏟}{x \cdot \frac{d (e^{y})}{d x} + e^{y} \cdot \frac{d x}{d x}}} = 1 - \frac{d y}{d x}

x \cdot \underset{CHAIN RULE}{\underset{⏟}{\frac{d (e^{y})}{d y} \cdot \frac{d y}{d x}}} + e^{y} \cdot 1 = 1 - \frac{d y}{d x}

x \cdot e^{y} \cdot \frac{d y}{d x} + e^{y} = 1 - \frac{d y}{d x}

Add

\frac{d y}{d x}

on both sides

x e^{y} \cdot \frac{d y}{d x} + e^{y} + \frac{d y}{d x} = 1

Subtract

e^{y}

from both sides

x e^{y} \cdot \frac{d y}{d x} + \frac{d y}{d x} = 1 - e^{y}

Take

\frac{d y}{d x}

common from the two terms in LHS

(x e^{y} + 1) \cdot \frac{d y}{d x} = 1 - e^{y}

Divides both sides by

x e^{y} + 1

\frac{d y}{d x} = \frac{1 - e^{y}}{x e^{y} + 1}

Result:

d

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