x22y =y2+ y' =?O A.y'22O B. y'=OC.y2y'O D. y'81ミ

We have to find y' for the given function: 2xy=x2y2 If we have too many expressions are in…

Answered: HI 2 y = y' =? %3D y2 O A. x2 y' = %3D…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

x2
2y =
y2
+ y' =?
O A.
y'
22
O B. y'
=
OC.
y2
y'
O D. y'
81ミ

Expert Solution

Step 1

We have to find $y'$ for the given function:

$2^{\frac{x}{y}} = \frac{x^{2}}{y^{2}}$

If we have too many expressions are in multiplication and division then we use logarithmic to simplify them

so taking log both side,

$\begin{array}{rcl} \ln (2^{\frac{x}{y}}) & = & \ln (\frac{x^{2}}{y^{2}}) \\ \frac{x}{y} \ln (2) & = & \ln (x^{2}) - \ln (y^{2}) \\ \frac{x}{y} \ln (2) & = & 2 \ln (x) - 2 \ln (y) . . . . . . . . . . (1) \end{array}$

Here we have used following logarithmic relation:

$\begin{array}{rcl} \ln (a^{m}) & = & m \ln (a) \\ \ln (\frac{a}{b}) & = & \ln (a) - \ln (b) \\ \ln (a b) & = & \ln (a) + \ln (b) \end{array}$

Step 2

We know the formula for derivatives:

$\begin{array}{rcl} \frac{d x^{n}}{d x} & = & n x^{n - 1} \\ \frac{d y}{d x} & = & y' \\ \frac{d \ln (x)}{d x} & = & \frac{1}{x} \\ \frac{d \ln (y)}{d x} & = & \frac{1}{y} \frac{d y}{d x} \\ \frac{d \frac{u}{v}}{d x} & = & \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}} \end{array}$

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