Calculus Help Please

Consider the implicit function

ln(xy)-cos^-1(x-1)=x

a. Find dy/dx.
b. Find y when x = 1.
c. Find the equation of the tangent line in slope-intercept form when x = 1.

Show steps please! Thanks!

asked by Tenshi
  1. ln(xy) - cos^-1(x-1) = x
    1/(xy) (y+xy') + 1/√(1-(x-1)^2) = 1
    Now just solve for y'. You should get

    y' = y(1 - 1/x - 1/√(2x-x^2))

    posted by Steve
  2. when x=1,
    ln(y) - pi/2 = 1
    y = e^(1+pi/2)

    at x=1, y' = -y

    so, now you have a point and a slope.

    y-e^(1+pi/2) = -e^(1+pi/2) (x-1)
    and convert that to slope-intercept.

    posted by Steve
  3. ln x y = lnx + ln y

    ln x + ln y -cos^-1(x-1) = x

    1/x + (1/y) dy/dx+(1/[1-(x-1)^2])^.5 = 1

    1/x +(1/y)dy/dx+(1/(-x^2+2x)^.5) = 1

    (1/y) dy/dx = 1 - 1/x - [1/(x(2-x))^.5

    dy/dx = y { 1 - 1/x - [1/(x(2-x))^.5 }


    when x = 1
    ln x + ln y -cos^-1(x-1) = x
    0 + ln y - pi/2 = 1
    ln y = 1 + pi/2
    e^ln y = y = e^(1+pi/2)
    y = 13.08 approx

    slope = dy/dx call y = 13
    = 13{1 -1/1 - [1/1] = 11

    y = 11 x + b
    13 = 11 + b
    b = 2
    y = 11 x + 2

    posted by Damon
  4. oops --- y' = -y, so the tangent line is

    y-13 = -13(x-1)
    y = -13x+26

    as shown at

    http://www.wolframalpha.com/input/?i=plot+y+%3D+e^%28x%2Barccos%28x-1%29%29%2Fx%2C+y%3D-13x%2B26

    posted by Steve

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