# Online polar derivative formulas tutor

Introduction:

• Incalculus, differentiation is the process of finding the derivative which means measuring how a function changes with respect to its input.
• Polar coordinate is a two dimensional coordinate system and it is defined by x = rcos θ, y = rsin θ.

Polar derivative formula:

Studentscan learn easily about polar derivative formulas from the online tutor.The online math tutors conduct the regular test for the students to improve their knowledge about polar derivative formulas. Following is the example and practice problems which show how the online tutor helps you for learning polar derivative formulas.

## Learning polar derivative formulas with example problems from the online tutor:

Derivative Examples:

Find derivative of the function r = 7 + 3sin θ

Step 1: Given function

r = 7 + 3sin θ

Step 2: Differentiate the given function r = 7 + 3sin θ with respect to ' θ '

`(dr)/(d theta)` = 3cos θ

Step 3: Substitute all values in the polar coordinate formula

`(dy)/dx` = `((dr)/(d theta) sin theta + r cos theta)/((dr)/(d theta)cos theta - r sin theta)` .

= `((3cos theta) sin theta + (7+ 3sin theta) cos theta)/((3cos theta) cos theta - (7 + 3sin theta)sin theta)`

= `(3cos theta sin theta + 7cos theta + 3sin theta cos theta)/(3cos^2 theta - 7sin theta - 3sin^2 theta)`

=  `(6cos theta sin theta + 7cos theta)/(3cos^2 theta - 7sin theta - 3sin^2 theta)`

Example problem 2:

Find derivative of the function r = 4 + 5cos θ

Step 1: Given function

r = 4 + 5cos θ

Step 2: Differentiate the given function r = 4 + 5cos θ with respect to ' θ '

`(dr)/(d theta)` = - 5sin θ

Step 3: Substitute all values in the polar coordinate formula

`(dy)/dx` = `((dr)/(d theta) sin theta + r cos theta)/((dr)/(d theta)cos theta - r sin theta)` .

= `((- 5sin theta) sin theta + (4 + 5cos theta) cos theta)/((- 5sin theta) cos theta - (4 + 5cos theta)sin theta)`

= `(- 5sin^2 theta + 4cos theta + 5cos^2 theta)/(- 5sin theta cos theta - 4sin theta - 5cos theta sin theta)`

= `(- 5sin^2 theta + 4cos theta + 5cos^2 theta)/(- 10sin theta cos theta - 4sin theta)`

Example problem 3:

Find derivative of the function r = tan 5θ

Step 1: Given function

r = tan 5θ

Step 2: Differentiate the given function r = tan 5θ with respect to ' θ '

`(dr)/(d theta)` = 5sec2

Step 3: Substitute all values in the polar coordinate formula

`(dy)/dx` = `((dr)/(d theta) sin theta + r cos theta)/((dr)/(d theta)cos theta - r sin theta)` .

=  `((5sec^2 5theta) sin theta + (tan 5theta) cos theta)/((5sec^2 5theta) cos theta - (tan 5theta)sin theta)`

= `(5sec^2 5theta sin theta + tan 5theta cos theta)/(5sec^2 5theta cos theta - tan 5thetasin theta)`

## Learning polar derivative formulas with practice problems from the online tutor:

1) Find derivative of the function r = 4 - 7sin θ

2) Find derivative of the function r = 2 + 3cos θ

3) Find derivative of the function r = tan 8θ

Solutions:

1) `(-14cos theta sin theta + 4cos theta)/(- 7cos^2 theta - 4sin theta - 7sin^2 theta)`

2) `(- 3sin^2 theta + 2cos theta + 3cos^2 theta)/(- 6sin theta cos theta - 2sin theta)`

3) `(8sec^2 8theta sin theta + tan 8theta cos theta)/(8sec^2 8theta cos theta - tan 8theta sin theta)`