Question

what is the third derivative of x/√1-x^2

Answer 1

y = x/√(1-x²)
Apply the rule for differentiation of u/v:

    dy/dx = [ √(1-x²) * 1 - x * {-2x/2√(1-x²) } ] / (1-x²)
              = [1 - x² + x² ] /(1-x²)³/²
             = 1/(1-x²)³/²

   d²y/dx² = -3/2 * 1/(1-x²)⁵/² * (-2x)
               = 3x /(1-x²)⁵/²

   d³y/dx³ = 3 * [(1-x²)⁵/² - x * 5/2 * (1-x²)³/² (-2x) ] / (1-x²)⁵
               = 3 * (1-x²)³/² * [ 1 - x² + 5 x² ] / (1-x²)⁵
               = 3 (1 + 4 x²) / ( 1 - x²)⁷/²

Answer 2

Here's your answer friend,
Refer to the attachment.
Hope it's useful and helpful!!!
@#shakti37