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?

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

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

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