Taking Derivatives of Parametric Systems
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
It's that simple.
This process works for any amount of variables.
Slope of Parametric Equations
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 Equations
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 :