# Parametric Equations Differentiation

[►Refer to Khan academy: Parametric equations differentiation](https://www.khanacademy.org/math/ap-calculus-bc/bc-advanced-functions-new/bc-9-1/v/parametric-function-differentiation) [`►Jump over to have some practice at Khan academy.`](https://www.khanacademy.org/math/ap-calculus-bc/bc-advanced-functions-new/bc-9-1/e/differentiate-parametric-functions)

## Why can we operate `dy/dx` algebraically?

[Refer to Khan academy: Addressing treating differentials algebraically](https://www.khanacademy.org/math/ap-calculus-bc/bc-diff-equations/modal/v/addressing-treating-differentials-algebraically)

> [Jump back to previous note: How to understand `dy/dx`](https://github.com/solomonxie/solomonxie.github.io/issues/49#issuecomment-389389887)

![image](https://user-images.githubusercontent.com/14041622/41278167-c1f8eace-6e5b-11e8-8a74-680783300059.png)

## Differentiate Parametric Equations

▼How to take derivative of a `parametric differential equation`? ![image](https://user-images.githubusercontent.com/14041622/42408083-09482802-81fa-11e8-81d2-ce60465673b4.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/42407942-99fc97a0-81f7-11e8-94ca-7dfc3787ee47.png) Solve:

* Let's do some trick to `dy/dt` and use this one instead:

  ![image](https://user-images.githubusercontent.com/14041622/42408123-2d156d4e-81fa-11e8-9d9e-296e67feefd6.png)
* Substitute the derivatives back:

  ![image](https://user-images.githubusercontent.com/14041622/42407997-b7a531da-81f8-11e8-90cc-93bcf4546e31.png)
* Plug in `t=1` to get the answer:

  ![image](https://user-images.githubusercontent.com/14041622/42408000-c3f3e332-81f8-11e8-847a-1247c66bedd1.png)

## Second derivatives of Parametric functions

![image](https://user-images.githubusercontent.com/14041622/42408348-de90aa86-81fd-11e8-995c-65569a529746.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/42408184-065424f6-81fb-11e8-91fd-c5f18248e545.png) Solve:

* Follow the rule, we got the first derivative:

  ![image](https://user-images.githubusercontent.com/14041622/42408217-a701cfe8-81fb-11e8-820c-fddeaeb561f7.png)
* And let's apply the rule for second derivative:

  ![image](https://user-images.githubusercontent.com/14041622/42408227-cb01d0be-81fb-11e8-96fd-aa66b57a154b.png)