Integral Some Functions

Before you see this post,

see

e ( mathematical constant )

Differential of A Function : sin x, cos x

Relationship between Integral and Differential

Introduction to Differential Equation

Differential Equations

1. Type

Ordinary Differential Equations

ex ) y" + y' = 0 , y" + 7 x( y' ) ^3 = 3x , ...

Partial Differential Equations

ex )

, ...

2. Linearity

Linear Differential Equations 

ex) y" + xy' = 7x,  8y"' + 3y' = xy, ...

Nonlinear Differential Equations

ex) (y"")^19 + (y")^ 30 - xy = 30x, ...

3. Order

1st, 2nd, 3rd, ... , n-th Order Differential Equations

ex)

y' + y^3 = 0 -> 1st order differential equation

(x^50)y" + (y')^300 + y = x -> 2nd order differential equation

x(y"')^20 + y = 3x^398 -> 3rd order differential equation

Differential of A Function : sin x, cos x

Before you see this post,

see

A Property of Limit of A Function : f(x)/g(x)

A Property of Limit of A Function : f(x)g(x)

Synthesis of Trigonometric Functions

Relationship between Integral and Differential

Before you see this post,

see

Riemann Sum and Riemann Integral

and

Error Estimate of Riemann Sum

f(q) can be integrated by Riemann integral and is a continuous function

and

F(x) can be differentiated.

Integral is an inverse operation of Differential.

Differential Function : [f(x)]^r = r[f(x)]^(r-1)

Before you see this post,

see

Differential Function : ln x

and

Differential of A Function : f(x)g(x)

q isn't zero.

Differential of A Function : f(x) / g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)

f(x) and g(x) are continuous functions.

Differential of A Function : f(x)g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)

f(x) and g(x) are a continuous functions

Differential of A Function : f(x)+g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)+g(x)

f(x) and g(x) are a continuous functions.

Derivative

Total differential

Before you see this post,

see

Derivative

A Property of Limit of A Function : f(x)g(x)

[pic 1]

Δf can be notated like [pic 1].

[pic 2]

We can continuously notate the function like [pic 2].

[pic 3]

Δf can be notated like [pic 3].

[pic 4]

Limit of Δf can be written to df and notated like [pic 4].

[pic 5]

[pic 6]

Let's use definition of partial differential in [pic 4]

and

a property of limit of a function like [pic 6].

See

A Property of Limit of A Function : f(x)g(x)

[pic 7]

We use definition in [pic 5] and a property in [pic 6] at a formula in [pic 4],

we can make a formula in [pic 7].

That is the formula of total differential.