Calc w/ Parametric Curves HELP?

Question

I keep getting -1 as my answer, but that answer is not one of the possible options... Please show your work on how to solve. Thank you!!

Answer 1

x = 5sint

Apply derivative to each side.

dx /dt = d/dt (5sint) = 5cost.

y = 5cost

Apply derivative to each side.

dy /dt = d/dt (5cost ) = -5sint.

dy / dx = (dy /dt ) / (dx /dt )

           = -5sint / 5cost

           = - tant

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x = 5sint

sint =x/5

t = sin^-1 (x/5 )

dy / dx =- tan(sin^-1 (x/5 ))

               =-(x/5)/√(1-(x/5)²)                                          (Since tan(sin^-1 (x)) =x/√ (1-x²))

                  =-x/√(25-x²)

Take the derivative both sides

d²y / dx²  =-(d/dx)(x/√(25-x²))

                 =-(√(25-x²)(1)-x(-1/(2√(25-x²))(2x))/(25-x²)

                 =-(√(25-x²)-x²(-1/(√(25-x²))/(25-x²)

                  =-((25-x²)+x²)/(25-x²)^3/2

                     =-25/(25-x²)^3/2

At t = 3π/4

x= 5sin3π/4 =5/√2

d²y / dx²  =-25/(25-x²)^3/2

                    =-25/(25-(5/√2)²)^3/2

                     =-25/(25-(25/2))^3/2

                      =-25/(25/2)^3/2

                       =-25*2^3/2/125

                 =-2√2/5

Option (c) is correct.