   Chapter 11.3, Problem 36E ### 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

# Write the equation of the line tangent to the curve x  ln  y +   2 x y =   2 at the point (1, 1).

To determine

To calculate: The equation of the tangent line to the curve xlny+2xy=2 at (1,1).

Explanation

Given Information:

The provided equation of the curve is, xlny+2xy=2 and point is (1,1).

Formula used:

The slope of tangent to a curve y=f(x) at point (x,y) is given by the derivative of the curve at that point.

The equation of a line passing through points (x1,y1) and slope m is given by,

yy1=m(xx1)

Calculation:

Consider the provided equation of curve,

xlny+2xy=2

Now, find the derivative dydx from xlny+2xy=2 by taking the derivative term by term on both sides of the equation as,

ddx(xlny)+ddx(2xy)=ddx(2)xdydx+lnydxdx+2xdydx+2ydxdx=0xyy+lny+2xy+2y=0

Solve for dydx as,

y(xy+2x)=lny2yy=ylny+2y2x+2xy

Now, the provided point is (1,1)

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