How do you differentiate parametric equations?

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Parametric equations are a set of equations which both depend on the same variable, such as t. An example of this would be:

x = 2t2​+1 and y = t​4​-2

As the value of t changes the equations will give you seperate values for x and for y which can be plotted on a coordinate grid.

To differentiate a parametric equation you must first differentiate both the equation for x and for y seperately with respect to t. So in this case it would be:

dx/dt = 4t and dy/dt = 4t3

We now have dx/dt and dy/dt. By simply divding dy/dt by dx/dt we get dy/dx as the dt cancels in the division (Since dividing is the same as multiplying by the reciprocal so (dy/dt)/(dx/dt) = (dy/dt)x(dt/dx) = dy/dx).

So for our example:

(dy/dt)/(dx/dt) = 4t3​/4t = t2 = dy/dx.

Timothy W. GCSE Maths tutor, A Level Maths tutor

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