# Derivatives

Constant - ex: 40.

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfRXqLTU6yfOGZnf2dE%2F-LfQNVuqtJHTw6LMWNot%2Fimage.png?alt=media\&token=5afe62a0-7293-4f0f-9e7e-d7876dcc425c)

Calculu's job is:

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfRXqLTU6yfOGZnf2dE%2F-LfQOFEYhifABG2I2Fw1%2Fimage.png?alt=media\&token=ceadc871-7d99-4629-bb1e-6f25a52815d4)

Given one of those functions, find the other. From function 1 to 2, that is called differential calculus. Going from 2 to 1 (take the speed and understanding what is the distances), that is called integral calculus.

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfRXqLTU6yfOGZnf2dE%2F-LfQSEG-kXYBMWpBDXJj%2Fimage.png?alt=media\&token=6370fc61-b104-4473-a1cc-9885d5ef5503)

### Big Picture Derivatives

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbPL4qkB8NvjQ0uec7%2F-LfbU46lv9JQtrPJgqjZ%2Fimage.png?alt=media\&token=c34aea74-6027-4ed0-af1b-7be8bc7f6d71)

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbPL4qkB8NvjQ0uec7%2F-LfbU_DmyBJQnkDxuOel%2Fimage.png?alt=media\&token=17501d9a-b113-4d79-987a-bea3e5118b6d)

Delta y is the difference between the heigh at point x and the height at point x + delta X. 

After simplifiying we get y'=2x+Deltax. But Delta x tends toward 0 because in the derivative we want smaller and smaller numbers. 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbPL4qkB8NvjQ0uec7%2F-LfbVAfe2EP-u7ahle0K%2Fimage.png?alt=media\&token=c81d86d0-c8bc-4b3a-8068-bdb48b804e45)

Therefore dy/dx = 2x. 

So now let's draw the slope: 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbPL4qkB8NvjQ0uec7%2F-LfbVgP4Pg79XlA4bXVk%2Fimage.png?alt=media\&token=e3ce1f9f-9192-441a-af90-8b4899c581dc)

Another example: 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbPL4qkB8NvjQ0uec7%2F-LfbVwNr6JPKUxET7iEp%2Fimage.png?alt=media\&token=5b05ae74-c34e-4b4e-9ec3-0d4b7d956fe2)