   Chapter 2.6, Problem 3E

Chapter
Section
Textbook Problem
50 views

# (a) Find y′ by implicit differentiation.(b) Solve the equation explicitly for y and differentiate to get y′ in terms of x.(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). x + y = 1

To determine

To find:

y' by using implicit differentiation.

Solution: y'= -yx

Explanation

1) Formula:

i. Power rule:

d dxxn=  n xn-1

ii. Constant rule:

ddx(c)=0

2) Given:

x+ y=1

3) Calculation:

Differentiating the equation  x 1/2 + y 1/2=1 with respect to x,

By using power rule and Chain rule for y term,

12x-12+12y-12dydx=0

Solving for dy/dx, subtract entire x term from both side

12y-12dydx=-12x-12

Divide both sides by 12y-12

Therefore,

dydx= -12x-1212y-12

By canceling common factors

dydx=- x-12y-12

Now convert negative exponent to positive by flipping the term from top to bottom and vice versa,

dydx=- y12x12

By writing the exponent term to radical,

y'= -yx

Conclusion:

Therefore,

y'= -yx

(b)

To Solve:

the equation explicitly for y and then differentiate it to get y’

Solution:

y =1- x2

y'= -1x+1

To solve the given equation x  + y=1  for y

1) Calculation:

y=1- x

Squaring both sides,

y =(1- x)2, this is solution of y in terms of x

This can be simplify further by using (a - b)2= a2-2ab+b2

y =1-2x+x

Now differentiate above equation with respect to ‘x’

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### ssIn Problems 7-20, find each limit, if it exists.

Mathematical Applications for the Management, Life, and Social Sciences

#### Convert 3,400 feet to miles.

Trigonometry (MindTap Course List)

#### The third partial sum of is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 