# Higher Order Derivatives

* First order derivative: `f'(x)` or `dy/dx`, also called "First Derivative"
* Second order derivative: `f''(x)` or `d²y/dx²`, also called "Second Derivative"
* ...

## `Second derivatives`

The second derivative of a function is simply the derivative of the function's derivative.

Notation: Leibniz's notation for second derivative is: ![image](https://user-images.githubusercontent.com/14041622/40231289-2c34f3da-5acd-11e8-9348-2ba4fb04a2aa.png) ![image](https://user-images.githubusercontent.com/14041622/46133022-1c338980-c272-11e8-8261-fd032981e31e.png)

## `Second derivatives of implicit equations`

[`▶ Back to previous note on Implicit Differentiation`](https://github.com/solomonxie/solomonxie.github.io/issues/49#issuecomment-390174936) [`▶ Practice at Khan academy: Second derivatives (implicit equations)`](https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-6/e/second-derivatives-implicit-equations)

[`▶ Online calculator for 2nd Derivative of Implicit equations`](https://www.symbolab.com/solver/step-by-step/implicit%20/frac%7Bd^%7B2%7Dy%7D%7Bdx^%7B2%7D%7D%2C%20%20y^%7B4%7D-2x%3D14)

### Example

![image](https://user-images.githubusercontent.com/14041622/46079842-dbcd0080-c1ca-11e8-8f62-ce0cd0a8e4ed.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/46079996-40885b00-c1cb-11e8-9431-911463a2c433.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/42376012-2e2d845a-8150-11e8-9f8a-2a58368ac08a.png) Solve:

*   The first derivative is:

    ![image](https://user-images.githubusercontent.com/14041622/42376194-d570d8f2-8150-11e8-8a55-0c4074fdfb3f.png)
*   The second derivative is:

    ![image](https://user-images.githubusercontent.com/14041622/42376217-e8875f56-8150-11e8-9068-dcfa6f6895a0.png)
*   Plug in `y=1`, and get:

    ![image](https://user-images.githubusercontent.com/14041622/42376247-f744cace-8150-11e8-8e7b-2563ecc9a1ce.png)
