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Find dy / dx and d²y / dx² and evaluate at the point : x = 2t +1 , y = t² + t at t = 2.

pur-new-sol

x = 2t + 1

y = t² + t , t=2;

dy/dx = (dy/dt) / ( dx/dt)

dy/dt = 2t + 1 ( d(tⁿ)/dt = nt^n-1)

dx/dt = 2

dy/dx = (2t + 1) / 2

= (2 * 2 + 1) / 2 ( at t = 2)

= 5/2

d²y / dx² = d(dy/dx) / dx

d²y / dx² = d ((dy/dt) / ( dx / dt)) / dx

d²y / dx² = [d ((dy/dt) / ( dx / dt)) / dt ] / (dx / dt)

= (d ((2t + 1) / 2)/dt ) / 2

= 1/2

dy/dx = 5/2

d²y / dx²= 1/2

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