For Parametric Equations, we will start to work with our usual variables (or parameters) like *x* and *y*. However, now we will use these variables/parameters where their value is dependent on an extra third variable/parameter. Often we use *t*, for time, as this third parameter.

Parametric Equations can be used for situations where for example, the *x* coordinate and the *y* coordinate are both based on time, *t*. **Instead** of writing the coordinates like (x, y), they can be written as *x* and *y* both as **functions** of t, like (x(t), y(t)) each of these parameters still plot on a graph the same way (x, y) would, itâ€™s just the values of the *x* and *y* both depend on the value of *t*.

If we have a parametric equation, where we have the parameters that are differentiable to *t*, of x = x(t) and y = y(t). And dx/dt does not equal 0, then,

=

We have the parametric equation (x, y) = (t, t^{2} + 1)

Calculate using =

First, for Parametric equations, recognize that we have our *x* & *y* terms as functions of *t*; *x(t)* & *y(t)*.

(x, y) = (t, t^{2} + 1)

x(t) = t

y(t) = t^{2} + 1

Now, to calculate we use

So differentiate x(t) to get ^{dx}/_{dt}

And y(t) to get ^{dy}/_{dt}

dx

dt

= 1

dy

dt

= 2t

=

=

2t

1

= 2t

Differentiate the parametric equation (x, y) = (3t^{2} + 2t, sin(t))

using =

We have (x, y) = (3t^{2} + 2t, sin(t))

To calculate =

, first find

dy

dt

&

dy

dt

We have x(t) = 3t^{2} + 2t

^{dx}/_{dt} = (2)3t^{2-1} + 2t^{0}

and

y(t) = sin(t)

^{dy}/_{dt} = cos(t)

=

=

cos(t)

6t + 2

The example below show how to find the slope of a tangent by differentiating.

Below is another worked examples showing how to find the slope of a tangent by differentiating.

For a graph that gives us the position of something based off time, we can use a method of differentiation of that graph to calculate its speed (or velocity)

If we have the variables x = x(t) and y = y(t) that give the position of something with regards to the time, *t*, and these variable functions of *t* are differentiable of *t*, then,

Speed =

(^{dx}/_{dt})^{2} + (^{dy}/_{dt})^{2}

Below a worked example showing how to find the speed of an object at given location and time.

Below is a worked examples showing how to determine at which location an object has greater speed.