# Calculus/Parametric Differentiation

## Taking Derivatives of Parametric Systems[edit]

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

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

In general, we say that if

and then:

and

It's that simple.

This process works for any amount of variables.

## Slope of Parametric Equations[edit]

In the above process, x' has told us only the rate at which x is changing, not the rate for y, 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 t:

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

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

If there is a time when both x' and y' are 0, that point is called a singular point.

## Concavity of Parametric Equations[edit]

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 t, 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 t: