Taking Derivatives of Parametric Systems[edit | edit source]
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[edit | edit source]
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[edit | edit source]
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 :