   Chapter 2.6, Problem 9E

Chapter
Section
Textbook Problem

# Find d y / d x by implicit differentiation. x 2 x + y = y 2 + 1

To determine

To find:

dydx

by Implicit differentiation.

Explanation

1) Formula:

i. Power rule:

d xndx=  n xn-1

ii. Quotient rule:

d (f/g)dx=f'*g-f*g'g2

iii. Constant rule:

d(c)dx=0

2) Given:

x2x + y=y2+1

3) Calculations:

Use quotient rule for the term x2x + y, chain rule for y2, constant rule for 1.

ddxx2x + y =  ddx(y2)+ddx(1)ddxx2*x+y-x2ddx(x+y) (x+y)2=2y2-1dydx+0

2xx+y-x21+ dydx(x+y)2=2ydydx

By simplifying the numerator,2xx+y-x2-x2dydx(x+y)2=2ydydx

2xx+y-x2(x+y)2- x2dydxx+y2=2ydydx

By adding x2dydxx+y2 to both sides,2xx+y-x2(x+y)2=2ydydx +x2dydxx+y2

Factor out dydx term from right side

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