# Derivative and differential

## Let's do basic

`Derivative`\(衍生物或派生物\) is some extracted thing from another thing. It may be a feature or characteristic, it's an abstract model for describing one thing in a higher view.

`Delta`\($$\Delta$$\) is a duration expressing the difference between two dates, time or date-time. We can seemly see this as a distance between two values.

`Differential` is a descriptor for describing the difference between two things.

**Sometimes we can see** `Delta` **and** `Differential` **as the same thing.**

## Derivative notation

1. Original notation

   $$
   \lim_{\Delta x \rightarrow 0}{\frac{\Delta y}{\Delta x}}
   $$

2. Lagrange's notation

   $$
   f^\prime(x)
   $$

   $$
   y^\prime
   $$

3. Leibniz's notation

   $$
   \frac{df(x)}{dx}
   $$

   $$
   \frac{dy}{dx}
   $$

   $$d$$ means small change. **You can read it:** `differential in x` **over** `differential in y` **is** `the derivative of that function`**.**

4. Newton's notation

$$
\dot y
$$

## What is derivative?

If it really has some meaning, that must be describing the average change in `y` when `x` get changed by this formula: $$\lim_{\Delta x \rightarrow 0}{\frac{\Delta y}{\Delta x}}$$

In that formula, when we assume that $$\Delta x$$\(`the change in x`\) is very small, near to 0, then $$\Delta y$$\(`the change in y`\) changes correspondingly. In that case, when the change in x and y is very small, $$\frac{\Delta y}{\Delta x}$$ becomes a tending, indicated where y is about to going.

## Graph derivative

Let me assume if we got a point A in a function curve.

The derivative of that function at that point is the slope of the tangent line at that point in that curve.

## What function feature derivative described?

![](../../.gitbook/assets/negative_x^2.png)

The $$|y^\prime|$$ bigger on one point, the $$y$$ of that function changes more when $$x$$ increases. It seems the slop of the tangent line will steeper then.

The $$|y^\prime|$$ smaller on one point until 0, the $$y$$ of that function changes less when $$x$$ decreases. It seems the slop of the tangent line gentler then.

If $$y^\prime$$ is positive, $$y$$ will going up.

If $$y^\prime$$ is negative, $$y$$ will going down.

It is the same thing if you replace $$y^\prime$$ with $$f^\prime(x)$$.

## All in all

$$
f^\prime(x) = y^\prime = \frac{dy}{dx} = \lim_{\Delta x \rightarrow 0}{\frac{\Delta y}{\Delta x}} = \lim_{\Delta x \rightarrow 0}{\frac{f(x + \Delta x) -f(x)}{\Delta x}}
\\ \\
$$

$$
f^\prime(x_0) = f^\prime(x)|_{x=x_0}
$$

