What is differential derivative definite integral indefinite integral and what is the difference between them?

The similarities and differences of derivative, differential and indefinite integral
I'm a little confused about these meanings now. It's better to have an easy to understand answer to solve my confusion
It's better to have examples

1. Differential and derivative are indistinguishable in English, but derivative is derivative concept: differential is tiny increment, infinitesimal increment, DX, Dy are differential, ratio dy / DX is derivative, is quotient

Finding limit derivative differential indefinite integral
Y = ln radical [numerator (x-1) (X-2) denominator (x + 3) (x + 4)] to find the derivative of Y
Let f (x) = {ax + 1, X be less than or equal to 2}
{xsquare + B, x > 2}
It can be derived at x = 2, and the values of constants A and B can be obtained
Let f (x) = {AE to the power of 2x, X}

1. Find the derivative y = ln [(x-1) (X-2) / (x + 3) (x + 4)]
y=ln(x-1)+ln(x-2)-ln(x+3)-ln(x+4)
So y ′ = 1 / (x-1) + 1 / (X-2) - 1 / (x + 3) - 1 / (x-4)
2. Let f (x) = ax + 1, when x ≤ 2; f (x) = x & # 178; + B, when x > 2}; be differentiable at x = 2, and find the values of constants A and B
If it is differentiable at x = 2, then it must be continuous at x = 2, so f (2) = 2A + 1 = 4 + B, that is, 2a-b = 3. (1)
If it is differentiable at x = 2, then the left derivative at x = 2 must be equal to its right derivative, so f '(2) = a = 4, and substituting (1) gives B = 5
3. Find the derivative y = 1 + X / √ (1-x)
y′=[√(1-x)+x/2√(1-x)]/(1-x)=(2-x)/[2(1-x)√(1-x)]
4. Find the limit x → 0lim [(1 / x) - 1 / ln (1 + x)]
x→0lim[(1/x)-1/ln(1+x)]=x→0lim{[ln(1+x)-x]/[xln(1+x)]}
=x→0lim{[1/(1+x)-1]/[ln(1+x)+x/(1+x)]}
=x→0lim{(-x)/[(1+x)ln(1+x)+1]}=0
5 find indefinite integral ∫ {1 / [(COS & # 178;) x]} D (COS x)
The original formula = - 1 / cosx + C
6 finding indefinite integral ∫ Sin & # 178; (x / 2) DX
The original formula = ∫ [(1-cosx) / 2] DX = (1 / 2) (x-sinx) + C
7.∫cos2x/(cosx-sinx)dx=∫[(cos²x-sin²x)/(cosx-sinx)]dx=∫(cosx+sinx)dx=sinx-cosx+C
8.∫(cotx/√sinx)dx=∫(cosx/sinx√sinx)dx=∫[(sinx)^(-3/2)]dsinx=-2/√sinx+C
9.∫dx/√(a²-x²)=∫d(x/a)/√[1-(x/a)²]=arcsin(x/a)+C
10.∫dx/(a²+x²)=∫d(x/a)/[1+(x/a)²]=arctan(x/a)+C
11.∫x²f(x³)f′(x³)dx=(1/3)∫f(x³)f′(x³)dx³==(1/3)∫f(x³)df′(x³)=(1/3)[f²(x³)]/2+C
12.∫tanx(tanx+1)dx=∫tan²xdx+∫tanxdx=∫[(1/cos²x)-1]dx-∫d(cosx)/cosx=tanx-x-ln︱cosx︱+C
13.∫[1/(1-x²)^(3/2]dx
Let x = sinu, then DX = cosudu, then the original formula = ∫ cosudu / cos & # 179; u = ∫ Du / cos & # 178; u = TANU + C = x / √ (1-x & # 178;) + C
14.∫[1/(x²+x-2)]dx=∫[1/(x+2)(x-1)]dx=(1/3)∫[1/(x-1)-1/(x+2)]dx=(1/3)[ln(x-1)-ln(x+2)]+C
=(1/3)ln[(x-1)/(x+2)]+C
15.∫(x²arctanx)/(1+x²)dx=∫x²d(arctanx)=x²arctanx-2∫xarctanxdx
Let arctanx = u, then x = TANU, DX = Du / cos & # u, then
-2∫xarctanxdx=-2∫(utanu/cos²u)du=-2∫(usinu/cos³u)du=-∫ud(1/cos²u)=-u/cos²u+∫du/cos²u
=-u/cos²u+tanu=-(1+x²)arctanx+x
So. ∫ (X & # 178; arctanx) / (1 + X & # 178;) DX = x & # 178; arctanx - (1 + X & # 178;) arctanx + X + C
16. Let ∫ XF (x) DX = arcsinx + C, find ∫ [1 / F (x)] DX
∫ XF (x) DX = arcsinx + C, ∫ XF (x) = (arcsinx + C) ′ = 1 / √ (1-x & # 178;), then f (x) = 1 / X √ (1-x & # 178;),
So ∫ [1 / F (x)] DX = ∫ x √ (1-x & # 178;) DX = - (1 / 2) ∫ [√ (1-x & # 178;)] d (1-x & # 178;) = - √ (1-x & # 178;) + C

The volume of an apple is about 250 (). A cubic centimeter B cubic decimeter C cubic meter
1. Uncle Wang enclosed a rectangular vegetable field with 20 1-meter-long wooden strips. How to maximize the enclosed area? How many square meters is the maximum area?
-------|----------------------
Length / M|
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Width / M|
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Area / m2|
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2. A cone-shaped wheat pile has a circumference of 12.56 meters and 2 meters at the bottom. If it is installed in a cylindrical grain bin with a diameter of 4 meters at the bottom, how high can it be piled? (formula)
The area of a circle is equal to that of a square. Then the far perimeter must be () the perimeter of the square.
A. Greater than B. less than C. equal to

The volume of an apple is about 250 (a cubic centimeter)
1. The square with a side length of 20 / 4 = 5 meters has the largest area,
The largest area is: 5 * 5 = 25 square meters
2. Cone bottom radius: 12.56 / (3.14 * 2) = 2
Radius of cylinder bottom: 4 / 2 = 2
How high can you stack: (1 / 3 * 3.14 * 2 * 2 * 1.2) / (3.14 * 2 * 2) = 0.4 (m)

The passenger car and the freight car run from a and B at the same time, and meet at a distance of 6 km from the midpoint. It is known that the speed of the freight car is 45 times that of the passenger car. How many kilometers are there between a and B?

A: the distance between a and B is 108 km

What is 1 cube + 2 cube + 3 cube + 4 cube + 5 cube + 6 cube + 7 cube

1 cubic + 2 cubic + 3 cubic + 4 cubic + 5 cubic + 6 cubic + 7 cubic = 28 cubic

X-1.8 = 4 X-2 = 15 3 + x = 45 5x = 30 according to the principle of balance
According to the principle of balance

X -1.8=4
Add 1.8 on both sides
X-1.8 1.8=4 1.8
X=5.8
X - 2=15
Add two on both sides
X-2 2=15 2
X = 17 3 x = 45 subtract 3 from both sides
3 x-3 = 45-3 x = 42 5x = 30 divide both sides by 5 5x △ 5 = 30 △ 5 x = 6 pure hand

As shown in the figure, the height of a cylinder is 8 cm. If its height is increased by 2 cm, its surface area will increase by 25.12 square cm. The volume of the original cylinder is 2 cm______ Cubic centimeter

The radius of the bottom circle of the cylinder: 25.12 △ 2 △ 3.14 △ 2 = 2 (CM); the volume of the original cylinder: 3.14 × 22 × 8 = 100.48 (cm3); answer: the volume of the original cylinder is 100.48 cm3. So the answer is: 100.48

A × 125% = B × 80% = C × 100% (a, B, C are not equal to 0), where the largest number is The smallest number is ．

Let the product of these three formulas be a, a = a △ 125%, 125% greater than 1, so their quotient is less than a, B = a △ 80%, 80% less than 1, so their quotient is greater than a, C = a △ 100%, 100% equal to 1, so their quotient is equal to A. so the largest number is B, and the smallest number is a. so the answer is: B, a

A British high school math problem, should be very simple
a fish weighs 1 pound plus half its weight.how much does it weigh altogether?

A fish weighs one pound - half its weight
x=1-x/2
Then x = 2 / 3 (LB)
A: it weighs two thirds of a pound

What is differential derivative definite integral indefinite integral and what is the difference between them?