Find the equation of all tangent lines having slope of -1 that are tangent to the curve y=(9)/(x+1)

y = -x + 5   and    y = -x - 7     (see attached graph)

Step-by-step explanation:

y =

  = 9(x + 1)⁻¹

Use the product rule to find the derivative

a = 9           a' = 0

b = (x + 1)⁻¹   b' = -(x + 1)⁻²

 ab' + a'b

= 9[-(x + 1)⁻²] + 0[(x + 1)⁻¹ ]

=

Set the derivative equal to the desired slope of -1 to solve for x

-1 =

-(x + 1)² = -9

 (x + 1)² = 9

 √(x + 1)² = √9  

    x + 1 = +/- 3

x + 1 = 3      x + 1 = -3

     x = 2          x = -4

Plug those values into the original equation to solve for y:

y =

  =

  = 3

(2, 3)

y =

  =

  = -3

(-4, -3)

Next, plug in the given slope (-1) and the coordinates above into the Point-Slope formula y - y₁ = m(x - x₁) to find the equations:

m = -1, (x₁ y₁) = (2, 3)                             m = -1, (x₁ y₁) = (-4, -3)

y - 3 = -1(x - 2)                                       y + 3 = -1(x + 4)

y - 3 = -x + 2                                          y + 3 = -x - 4

    y = -x + 5                                                y = -x - 7

step-by-step explanation:

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