The second derivative of a parametric curve is given by which formula?

The second derivative of a parametric curve is given by which formula?

• A. d^2y/dx^2 = (d/dt)(dy/dx) / (dx/dt)
• B. d^2y/dx^2 = (dx/dt) / (dy/dt)
• C. d^2y/dx^2 = (d/dt)(dx/dy) / (dy/dt)
• D. d^2y/dx^2 = (d^2y/dt^2) / (dx/dt)^2

Answer: (A)

When x and y depend on a parameter t, the slope is dy/dx = (dy/dt)/(dx/dt). The second derivative with respect to x measures how this slope changes as x changes, so you differentiate dy/dx with respect to t and then divide by dx/dt (chain rule: d/dx = (1/(dx/dt)) d/dt). This gives d^2y/dx^2 = [d/dt (dy/dx)] / (dx/dt). If you expand that, you’d get an explicit form: d^2y/dx^2 = [ (d^2y/dt^2)(dx/dt) − (dy/dt)(d^2x/dt^2) ] / (dx/dt)^3, which confirms the compact form. The other expressions don’t match the derivative with respect to x for a parametric curve.

When x and y depend on a parameter t, the slope is dy/dx = (dy/dt)/(dx/dt). The second derivative with respect to x measures how this slope changes as x changes, so you differentiate dy/dx with respect to t and then divide by dx/dt (chain rule: d/dx = (1/(dx/dt)) d/dt). This gives d^2y/dx^2 = [d/dt (dy/dx)] / (dx/dt).

If you expand that, you’d get an explicit form: d^2y/dx^2 = [ (d^2y/dt^2)(dx/dt) − (dy/dt)(d^2x/dt^2) ] / (dx/dt)^3, which confirms the compact form. The other expressions don’t match the derivative with respect to x for a parametric curve.