Parametrics – Curves, Calculus, Polar Coordinates  (continued)

 

 

Calculate the derivative of a function expressed in parametric form.

 

Let               y  =  y(x)  where   x  =  x(t)      Then     y  =  y ( x(t) )

 

 

 

 

Strategy:    Use the “chain rule”    to calculate  dy/dx .

 

 

Application of the chain rule gives         dy/dt  =  dy/dx   dx/dt

 

Then the first derivative is         dy/dx  =   [dy/dt]  /  [dx/dt]      provided that   dx/dt    0.

 

 

Higher derivatives may also be calculated for parametric representation of functions.

 

For the second derivative    d2y/dx2  =  d/dx [dy/dx]. 

 

So replace   y with   dy/dx     in      dy/dx  =  [dy/dt] / [dx/dt]  

 

This substitution yields

 

                         d2y/dx2  =  d/dt [dy/dx] / [dx/dt]       for the second derivative.

 

Click here for examples of derivatives of a function in parametric form.

 

 

Sometimes problems in calculus may be more easily represented using a coordinate

system other than cartesian coordinates (x,y).  One such system is the polar coordinate

system.  The figure below shows the graphical relation between cartesian coordinates (x,y)

and polar coordinates  (r, θ) .

                                    

 

Here we see that       x  =  r cos θ    and    y  =  r sin θ

 

Click here for examples involving polar coordinates.