   Chapter 2.6, Problem 23E

Chapter
Section
Textbook Problem

# Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find d x / d y . x 4 y 2 − x 3 y + 2 x y 3 = 0

To determine

To find:

dxdy

By using implicit differentiation.

Solution:

dxdy=-2x4y+x3-6xy24x3y2-3x2y+2y3

Explanation

1) Formula:

i. Constant Multiplication:

ddxcf=cddxf

ii. Power Rule:

ddxxn=nxn-1

iii. Product Rule:

ddxf*g=fddxg+gddxf

2) Given:

x4y2-x3y+2xy3=0

3) Calculation:

x4y2-x3y+2xy3=0

Differentiate this equation with respect to y

ddyx4y2-x3y+2xy3=0

ddyx4y2-ddy(x3y)+ddy(2xy3)=0

By using constant multiple rule,

ddyx4y2-ddy(x3y)+2ddy(xy3)=0

Use product rule for differentiation

x4ddyy2+y2ddyx4-x3ddyy+yddy(x3)+2xddyy3+y3ddy(x)=0

Use Power rule combined with chain rule

x42y+y24x3dxdy-x3+y3x2dxdy+

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