   Chapter 2.6, Problem 12E

Chapter
Section
Textbook Problem

# Find d y / d x by implicit differentiation. cos ( x y ) = 1 + sin y

To determine

To find:

dydx

by Implicit differentiation.

Solution: dydx= -ysin(xy) cosy+xsin(xy)

Explanation

1) Formula:

i. Product rule:

d (f*g)dx=f'*g+f*g'

ii. Constant rule:

d(c)dx=0

2) Given:

cosxy=1+sin y

3) Calculations:

cosxy=1+sin y

Use chain rule for the term cos(xy), product rule on the term  xy, constant rule on 1.

ddxcos(xy )=  ddx(1+siny)-sin(xy)ddx(xy)= ddx1+ddxsiny

-sinxyddxx*y+ x*ddx y=0+cosydydx That is

-sinxyy+ xdydx=cosydydx

Distribute -sin(xy)

-y sinxy-xsin

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