   Chapter 11.6, Problem 8E 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.] e x y − y = 2

To determine

To calculate: The value of dydx for the equation exyy=2 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 exyy=2.

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) The derivative of function f(x)=un using the chain rule is

f(x)=ddx(un)=nun1dudx,

Where, u is the function of x.

3) The derivative of e raised to x is ddxex=ex.

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

f'(cx)=cf'(x)

Where, c is constant.

Calculation:

Consider the equation, exyy=2

Take ddx of both sides,

ddx(exyy)=ddx(2)ddx(exy)dydx=ddx(2)

Apply the product rule of derivative,

ddxexy+exdydxdydx=0exy+exdydxdydx=0

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

exdydxdydx=0exy=exy

Take out dydx as common,

(ex1)dydx=exy

Simplify the derivative through dividing both sides by (ex1),

(ex1)(ex1)dydx=exy(ex1)dydx=exyex1=exy1ex

Thus, the value of dydx for equation exyy=2 is exy1ex using the implicit differentiation

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