# Polar Curve Functions (Differential Calc))

[`▶ Practice at Khan academy.`](https://www.khanacademy.org/math/ap-calculus-bc/bc-advanced-functions-new/bc-9-7/e/differentiate-polar-functions)

[Refer to Khan academy: Polar functions derivatives](https://www.khanacademy.org/math/ap-calculus-bc/bc-advanced-functions-new/bc-9-7/v/derivatives-in-polar-coordinates)

> [`▶ Proceed to: Area of Polar Curves (Integral Calc)`](https://github.com/solomonxie/solomonxie.github.io/issues/49#issuecomment-398657472)

![image](https://user-images.githubusercontent.com/14041622/42408971-cec59e88-8205-11e8-9e14-bc1a910846da.png)

> In the `Polar World`, instead of the relationship between `y & x`, the function is now representing the relationship between `Radius & Angle`, which could be presented as:

![image](https://user-images.githubusercontent.com/14041622/42408955-8bb18e9a-8205-11e8-8750-67370131d552.png)

## `Finding the right boundaries`

> The most tricky part in Polar system, is finding the right boundaries for `θ`, and it will be the first step for polar integral as well.

## `Differentiate Polar Functions`

**Taking derivative of Polar function** is actually **DIFFERENTIATING PARAMETRIC FUNCTION**. To take the derivative we need to:

*   Convert the `Polar function` in terms of `x & y`:

    ![image](https://user-images.githubusercontent.com/14041622/42409085-7f99bb26-8207-11e8-9b8f-53fd99405428.png)
* Take derivative of the parametric function.

### Example

![image](https://user-images.githubusercontent.com/14041622/42408886-70192f54-8204-11e8-9eaf-ff0d0b4cc7b7.png) Solve:

*   Since it's asking for the `Rate of change of y-coordinate`, so we convert the polar function to `rectangular function`:

    ![image](https://user-images.githubusercontent.com/14041622/42409148-900f2008-8208-11e8-8433-6ec1dc53eaa2.png)
*   And we take the derivative `dy/dΘ`:

    ![image](https://user-images.githubusercontent.com/14041622/42409155-b66604e2-8208-11e8-9b4a-1cfd57b53964.png)
*   Plug in the point `Θ=π` and get:

    ![image](https://user-images.githubusercontent.com/14041622/42409162-ce8d4f58-8208-11e8-8bd5-a1b4062f71a4.png)

## `Tangents to Polar curves`

Steps:

* Find the slope `dy/dx`
* Convert the polar function to get the `x(θ)` and `y(θ)` parametric equations
* Solve `dy/dx` and get the slope
* Plug in the point's information to solve for `x & y`
* Get the equation of the line (tangent).

### Example

![image](https://user-images.githubusercontent.com/14041622/42409334-61386854-820b-11e8-84de-f5676a5543f4.png) Solve:

* To find the tangent line, we need to get the slope first, which is `dy/dx`.
*   And `dy/dx` would be a `parametric problem`:

    ![image](https://user-images.githubusercontent.com/14041622/42409359-d9da5704-820b-11e8-9dd3-6bcce56f25f4.png)
*   Plug in the Θ value, to evaluate the slope:

    ![image](https://user-images.githubusercontent.com/14041622/42409371-119c3dc4-820c-11e8-8eaf-8a0b6da1ca8f.png)
*   Find the `x & y` value according to the Θ:

    ![image](https://user-images.githubusercontent.com/14041622/42409376-38b1ac46-820c-11e8-91f4-cb088fcc8ed6.png)
*   Now we got everything to form the equation for the tangent line:

    ![image](https://user-images.githubusercontent.com/14041622/42409378-4c5659f4-820c-11e8-9675-dd1bc5d54907.png)

    ![image](https://user-images.githubusercontent.com/14041622/42409379-512ce470-820c-11e8-882c-d0265c077d41.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/42410082-d26ab8fa-8216-11e8-9362-9e88eb6f2a9f.png) Solve:

*   First, we need to convert the polar function to `x(θ) & y(θ)`:

    ![image](https://user-images.githubusercontent.com/14041622/42410205-4a6413a0-8218-11e8-8721-95acf2208c63.png)
*   And we need to find the slope `dy/dx`:

    ![image](https://user-images.githubusercontent.com/14041622/42410164-d9a5b8f8-8217-11e8-9bd5-1355d13c31ff.png)
*   Since it's a horizontal tangent, so `Slope =0`, which means `dy/dx =0`. But `dx` is dominator can't be zero, so we can set `dy = 0` and solve for θ:

    ![image](https://user-images.githubusercontent.com/14041622/42410215-74fd8092-8218-11e8-85c2-975a74be7fc7.png)
*   So the answer is:

    ![image](https://user-images.githubusercontent.com/14041622/42410218-940013c4-8218-11e8-8065-15bee832ee77.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/42410271-bcbc245a-8219-11e8-8f8d-0e8bd320f408.png) Solve:

* To find a vertical tangent, we have to set the dominator of the slope as 0, that's the only thing makes it **undefined**.
* The slope is `dy/dx`, so we set `dx = 0`.
*   The equation for `x` is:

    ![image](https://user-images.githubusercontent.com/14041622/42410294-0226d9c2-821a-11e8-91e3-cdf2809609b2.png)

    ....
