   Chapter 11.6, Problem 36E 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 31–42, use implicit differentiation to find (a) the slope of the tangent line and (b) the equation of the tangent line at the indicated point on the graph. (Round answers to four decimal places as needed.) If only the x-coordinate is given, you must also find the y-coordinate. [HINT: See Example 2 and 3.] ( x y ) 2 + x y − x = 8 , ( − 8 , 0 )

(a)

To determine

To calculate: The slope of the tangent line to the graph of the equation (xy)2+xyx=8 at the point (8,0) using the implicit differentiation.

Explanation

Given Information:

The equation is (xy)2+xyx=8 and the point is (8,0).

Formula used:

Slope of tangent to the graph of f(x) at point (x1,y1) is given by dydx|(x1,y1).

The derivative of function f(x)=un using the chain rule is f(x)=ddx(un)=nun1dudx, where u is the function of x.

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).

Calculation:

Consider the equation, (xy)2+xyx=8

Find the slope of tangent line to the equation (xy)2+xyx=8 by determining the derivative.

Then, take ddx of both sides of the equation,

ddx((xy)2+xyx)=ddx(8)ddx(xy)2+ddx(xy)ddx(x)=ddx(8)

Apply the chain rule and the product rule of derivative,

2(xy)21ddx(xy)+(dxdxy+xdydx)1=02xyddx(xy)+(1y+xdydx)1=02xyddx(xy)+(y+xdydx)1=0

Now, again apply the product rule,

2xy(dxdxy+xdydx)+(y+xdydx)1

(b)

To determine

To calculate: The equation of the tangent line to the graph of the equation (xy)2+xyx=8 at the point (8,0).

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