   Chapter 2.6, Problem 20E

Chapter
Section
Textbook Problem

# Find d y / d x by implicit differentiation. tan ( x − y ) = y 1 + x 2

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]

2) Given:

tanx - y=y1+x2

3) Calculations:

tanx - y=y1+x2

Differentiate it with respect to x,

Use chain rule on an(x - y), and quotient rule on right side,

ddx [tan(x - y) ] =  ddxy1 + x2

sec2(x-y)ddxx-y= 1 + x2ddxy - y *ddx (1+ x2 )[( 1+ x2)2]

sec2(x-y)ddxx-ddxy=1 + x2dydx - y * (0+2x )[( 1+ x2)2]

sec2(x-y)1-dydx=1 + x2dydx - 2xy [( 1+ x2)2]

Distribute sec2(x-y)) on left side

sec2x-y - sec2(x-y)dydx=1 + x2dydx - 2xy [

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