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To differentiate parametric equations, we must use the chain rule.
If x = 2at2 and y = 4at, find dy/dx
In this case, dx/dt = 4at and so dt/dx = 1/(4at)
Also dy/dt = 4a. Hence:
dy/dx = 4a × 1/4at = 1/t
Finding the Second Derivative
Finding the second derivative is a little trickier.
We use the fact that:
To find the second derivative in the above example, therefore:
d2y = d(1/t) × dt
dx2 dt dx
= -1 × 1 .