Differentiation Of Parametric Function





Differentiation of parametric function is another interesting method in the topic differentiation. In this method we will have two functions known as x and y. Each function will be defined using another third variable. To understand this topic more let us see some examples.

Example 1:

Find  dy/dx for the below parametric functions.

x = a cos θ y = b sin θ

Solution:

Here the two variables x and y are defined using third variable θ. By using the function x we can only find dx/dθ and by using the function y we can only find dy/dθ. By dividing these two derivatives we can find dy/dx.

x = a cos θ                       y = b sin θ

dx/dθ = a (-sin θ)             dy/dθ = b cos θ

dx/dθ = - a sin θ

Now we can find dy/dx

 dy/dx = (dy/dθ)/(dx/dθ)

 dy/dx = (b cos θ)/(- a sin θ)

 dy/dx = -b/a (cos θ/ sin θ)

 dy/dx = -(b/a) cot θ          differentiation of parametric function      differentiation of parametric function 


Example 2:

Find dy/dx by using the parametric functions

x = a sec³ θ    y = b tan³ θ

Solution:

Here the two variables x and y are defined using third variable θ. By using the function x we can only find dx/dθ and by using the function y we can only find dy/dθ. By dividing these two derivatives we can find dy/dx.

x = a sec³ θ

dx/dθ = 3a sec² θ(sec θ tan θ) 

         = 3a sec³ θ tan θ

y = b tan³ θ

dy/dθ = 3b tan² θ  sec² θ

Now we have divide those two derivatives to find dy/dx

     dy/dx = (dy/dθ)/(dx/dθ)

              = (3b tan² θ  sec² θ)/(3a sec³ θ tan θ)

              = (b/a) (tan θ/sec θ)

              = (b/a) (sin θ/cos θ)/(1/cos θ)

              = (b/a) (sin θ/cos θ) x (cos θ/1)

              = (b/a) sin θ 

Related Topics


Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

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