- 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:
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.
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.
Q. Find when and .
Use the chain rule:
Q. Let and , find at .
Put in .
Q. Find when and
Here the parameter given is r:
Now use the chain rule: