Summary
- When x and y are expressed in terms of a third variable it is called a parameter.
- To differentiate parametric equations to find derivative of y with respect to x, we use the chain rule method.
- When the parameter in the equations is “t”, the chain rule is defined as:
What is parameter?
We now completely understand that differentiation is the process that we use to find the gradient of a curve and it is denoted by , we call it a derivative or differential coefficient of y with respect to x.
Moving forward, we know that a relation between the coordinates x and y is known as ”cartesian form”. However, there are times when it is convenient to express x and y in terms of a third variable which is called a parameter.
Suppose, the equation of a curve is given by two parametric equations. For example, two parametric equations of a circle with centre zero and radius a are given by: and
here t is the parameter.
Chain Rule Method
Now to differentiate these parametric equations, we must understand that in each case we can’t differentiate with respect to x, hence we cannot directly find , we will use the chain rule here:
- For each of the above equations we will only be able to find
, and
and then using the chain rule we find
.
- The chain rule will be applied like this:
Let’s do an example now to see how we carry out parametric differentiation.
Example #1
Q. Find when
and
.
Solution:
and
Use the chain rule:
We know:
But since:
Therefore:
Ans
Example #2
Q. Let and
, find
at
.
Solution:
and
Put in
.
Hence:
Ans
Example #3
Q. Find when
and
Solution:
Here the parameter given is r:
and
Now use the chain rule:
We know:
But:
Ans
Reference
- https://studylib.net/doc/13474624/parametric-differentiation
- https://slideplayer.com/slide/10093157/