**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:

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.

#### 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/