   Chapter 11.3, Problem 50E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# (a) Find y' implicitly for x 3 −   y 3 =   8. (b) By taking derivatives implicitly, use part (a) to show that y " = 2 x ( y − x y ' ) y 3 (c) Substitute x 2 / y 2 for y in the expression for y" and simplify to show that y " = 2 x ( y 3 − x 3 ) y 5 (d) Does y " =   −   16 x / y 5 ? Why or why not?

(a)

To determine

To calculate: The value of y if x3y3=8.

Explanation

Given Information:

The provided equation is:

x3y3=8

Formula used:

The chain rule of derivative,

dydx=dydududx

Calculation:

Consider the provided equation:

x3y3=8

First, take the derivative of both sides of the equation with respect to x as:

ddx(x3y3)=ddx(8)

Now, use the chain rule:

dydx=dydududx

(b)

To determine

To prove: The y=2x(yxy)y3 if the value of y is x2y2.

(c)

To determine

To prove: The equation y=2x(y3x3)y5 if y=x2y2.

(d)

To determine

The validity of statement “y=16xy5” for x3y3=8.

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