For an equation written in its parametric form, the first derivative is
and the second derivative is
where and are the coordinates of the points described by the equation and is the parameter.
A brute-force method for finding the first derivative is to eliminate the parameter, but this makes problems involving the parameter more complex. More elegantly, consider the derivative of with respect to
By the chain rule,
The first term on the right side is the first derivative of the parametric function. In order to isolate it, divide both sides by
In practice, differentiate the parametric functions of and independently and plug them into the relation above.
If and what is when
First, compute the derivatives of and in terms of
The second derivative is, by definition,
By the chain rule, so
From above, so
Finally, evaluate on the right side by using the quotient rule:
This equation is less headache-inducing if written using Newton's dot notation, by which represents the first derivative of with respect to and represents the second derivative of with respect to .
If and what is the second derivative of in terms of
First, evaluate the first and second derivates needed:
Next, evaluate the second derivative:
The tangent equation represents a straight linear line that creates a right angle at the point of tangency. The formula of a line is described in Algebra section as "point-slope formula":
In parametric equations, finding the tangent requires the same method, but with calculus:
Tangent of a line is always defined to be the derivative of the line. Thus, for slope, we use
- Helsing, v. Animated involute of circle. Retrieved May 18, 2016, from https://commons.wikimedia.org/wiki/File:Animated_involute_of_circle.gif