   Chapter 11.6, Problem 5E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 1–10, find d y / d x , using implicit differentiation. In each case, compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative. [HINT: See Example 1.] 2 x + 3 y = x y

To determine

To calculate: The value of dydx for the equation 2x+3y=xy using the implicit differentiation. Also compare the result obtained by rearranging the equation for y as a function of x and taking the derivative.

Explanation

Given information:

The provided equation is 2x+3y=xy.

Formula used:

1) Product rule of derivative of differentiable functions, f(x) and g(x) is

ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x).

2) Constant multiple rule of derivative of function f(x) is

f'(cx)=cf'(x)

Where, c is constant.

3) Quotient rule of derivative of differentiable functions, f(x) and g(x) is

ddx[f(x)g(x)]=f'(x)g(x)f(x)g'(x)[g(x)]2

Where, g(x)0.

Calculation:

Consider the equation, 2x+3y=xy

Take ddx of both sides,

ddx(2x+3y)=ddx(xy)ddx(2x)+ddx(3y)=ddx(xy)

Apply constant multiple rule and the product rule of derivative,

2dxdx+3dydx=dxdxy+xdydx21+3dydx=1y+xdydx2+3dydx=y+xdydx

Evaluate the value of dydx by isolating it to the left side,

3dydxxdydx=y2

Take out dydx as common,

(3x)dydx=y2

Simplify the derivative through dividing both sides by (3x),

(3x)(3x)dydx=y2(3x)dydx=y23x

Thus, the value of dydx for equation 2x+3y=xy is y23x using the implicit differentiation

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