Parametric Differentiation

To differentiate parametric equations, we must use the chain rule.

Example

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:

Example

To find the second derivative in the above example, therefore:

d²y =  d(1/t) × dt
dx²    dt         dx

= -1 ×   1    .
   t²      4at