Introduction to solving parametric derivative formulas:

 The  derivative is a measure of how a function changes as its input changes.

 

 In derivative, if both x and y variables depend on the third independent variable ' t ' , it is called as parametric derivative.

 

It is easy for solving derivative of parametric function using parametric formulas. In this article, we are going to see the parametric derivative formulas and example problems with the solution, which help you to learn more about parametric derivative formulas.

 

 

 

Parametric derivative formulas:

 

Formula of first derivative parametric function:

                        `(dy)/(dx) = ((dy)/(dt))/((dx)/(dt))`

Formula of second derivative parametric function:

                       ` (d^2y)/(dx^2)` = `(d/(dt)((dy)/(dx)))/((dx)/(dt))`

                              = `((dx)/(dt)((d^2y)/(dt^2)) - (dy)/(dt)((d^2x)/(dt^2)))/((dx)/dt)^3` .

Following is the example problem of first derivative parametric function using parametric formulas.

 

Learn to solve parametric derivative formulas with example problems:

 

Solving parametric derivative formulas with example problem 1:

Find derivative of the parametric functions x = 17t4 + 5 and y = 2t2.

Solution:

      Step1: Given functions

                           x = 17t4 + 5

                           y = 2t2

      Step 2: Differentiate the function x = 17t4 + 5 with respect to ' t '

                           x = 17t4 + 5 

                           `(dx)/(dt)` = 68t3

      Step 3: Differentiate the function y = 2t2 with respect to ' t '

                           y = 2t2

                           `(dy)/(dt)` = 4t

      Step 4: Substitute the obtained values in parametric derivative formula to find ` (dy)/(dx)`

                           `(dy)/(dx) = ((dy)/(dt))/((dx)/(dt))`.

                          ` (dy)/(dx)``(4t)/(68t^3)`

                                 = `1/(17t^2)` .

Solving parametric derivative formulas with example problem 2:

Find derivative of the parametric functions x = 3t2 + 4t and y = 4t + 7

Solution:

      Step1: Given functions

                           x = 3t2 + 4t

                           y = 4t + 7

      Step 2: Differentiate the function x = 3t2 + 4t with respect to ' t '

                           x = 3t2 + 4t 

                           `(dx)/(dt)` = 6t + 4

      Step 3: Differentiate the function y = 4t + 7 with respect to ' t '

                           y = 4t + 7

                           `(dy)/(dt)` = 4

      Step 4: Substitute the obtained values in parametric derivative formula to find ` (dy)/(dx)`

                           `(dy)/(dx) = ((dy)/(dt))/((dx)/(dt))`.

                          ` (dy)/(dx)``4/(6t + 4)`

                                 =   `2/(3t + 2)`  .

 

Solving parametric derivative formulas with example problem 3:

Find derivative of the parametric functions x = 7sin 2t and y = 5cos 3t

Solution:

Step1: Given functions

                           x = 7sin 2t

                           y = 5cos 3t

Step 2: Differentiate the function x = 7sin 2t with respect to ' t '

                           x = 7sin 2t

                           `(dx)/(dt)` = 14cos 2t

Step 3: Differentiate the function y = 5cos 3t with respect to ' t '

                           y = 5cos 3t

                           `(dy)/(dt)` = - 15sin 3t

Step 4: Substitute the obtained values in parametric derivative formula to find ` (dy)/(dx)`

                           `(dy)/(dx) = ((dy)/(dt))/((dx)/(dt))`.

                          ` (dy)/(dx)``(-15sin 3t)/(14cos 2t)`