Calculus/Parametric Differentiation

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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   and solve.

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

If there is a time when both   and   are   , 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   :

 

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Parametric Differentiation