## Taking Derivatives of Parametric SystemsEdit

Just as we are able to differentiate functions of , we are able to differentiate and , which are functions of . Consider:

We would find the derivative of with respect to , and the derivative of with respect to :

In general, we say that if

and then:

and

It's that simple.

This process works for any amount of variables.

## Slope of Parametric EquationsEdit

In the above process, has told us only the rate at which is changing, not the rate for , and vice versa. Neither is the slope.

In order to find the slope, we need something of the form .

We can discover a way to do this by simple algebraic manipulation:

So, for the example in section 1, the slope at any time :

In order to find a vertical tangent line, set the horizontal change, or , equal to 0 and solve.

In order to find a horizontal tangent line, set the vertical change, or , equal to 0 and solve.

If there is a time when both are 0, that point is called a singular point.

## Concavity of Parametric EquationsEdit

Solving for the second derivative of a parametric equation can be more complex than it may seem at first glance.

When you have take the derivative of in terms of , you are left with :

.

By multiplying this expression by , we are able to solve for the second derivative of the parametric equation:

.

Thus, the concavity of a parametric equation can be described as:

So for the example in sections 1 and 2, the concavity at any time :