Derivative, differential, indefinite integral

Derivative, differential, indefinite integral

The x-th derivative of y can be obtained by logarithmic derivative method or as a composite function
x^y＋y^x＝3
e^(ylnx)＋e^(xlny)＝3
e^(ylnx)×[y'lnx＋y/x]＋e^(xlny)×[lny＋xy'/y]＝0
x^y×[y'lnx＋y/x]＋y^x×[lny＋xy'/y]＝0
Substituting x = 1, y = 2, we get y '= - 2-2ln2
Therefore, the tangent equation is Y-2 = - (22ln2) (x-1), that is, y = - (22ln2) x-2ln2 + 4

All the original functions of integrable function are called its indefinite integral
Is it right or wrong

Let f (x) be an original function of function f (x). We call all the original functions f (x) + C (C is any constant) of function f (x) indefinite integral
yes

In a cage, there are chickens and rabbits. Count their feet. There are 128 of them. Count their heads. There are 46 brothers. How many chickens and rabbits are there in the cage

There are x rabbits and Y chickens
x+y=46
4x+2y=128
x=18
y=28

.x+y=11,xy=18
There are two answers

The answer is x = 9, y = 2 or x = 2, y = 9. There are two answers. This is very normal, because according to the two formulas, we get a system of quadratic equations with one variable. Unless there is a special explanation for the question, we have to write down both answers

A number minus 6 out of 7, plus 1 out of 3, and the sum is 8 out of 21

8/21-1/3+6/7
=1/21+6/7
=19/21

A mathematics problem of grade two in junior high school y = x ^ 2-5x
y=x^2-5x
-2≤x≤7
Find x
And y

y=x²-5x+25/4-25/4
=(x-5/2)²-25/4
-2≤x≤7
-9/2≤x-5/2≤9/2
So 0 ≤ (X-5 / 2) & sup2; ≤ 81 / 4
-25/4≤(x-5/2)²-25/4≤14
therefore
-2≤x≤7
-25/4≤y≤14

In a tree planting activity, the total number of trees planted by the two groups is the same, which is more than 100. The two groups are known to admit defeat. In the first group, one person planted 5 trees, and the others planted 13 trees
In the second group, one person is worth 4 trees, and the others are worth 10 trees. How many people are there in the two groups?

According to the meaning of the topic, the number of trees in each group, divided by 13 more than 5; divided by 10 more than 4; Chinese remainder theorem problem

P (x): a square > 4B Q (x): x + ax + B is equal to zero with real roots What condition is p (x) Q (x)

The sufficient and necessary condition for x ^ 2 + ax + B = 0 to have a real root is as follows:
a^2-4b>=0;
a^2>=4b;
so:
A square > 4b is a sufficient and unnecessary condition

On Sunday, Xiao Ming and seven classmates, a total of 8 people, went on an outing. On the way, he spent 20 yuan to buy drinks. The store only had coke and milk tea. It is known that coke costs 2 yuan a cup and milk tea costs 3 yuan a cup. If 20 yuan is just used up. (1) how many ways to buy? How many cups of cola and milk tea in each way? (2) When each person has at least one drink and at least two milk teas, how many ways to buy them?

(1) Let's buy Coke and milk tea as X and Y cups respectively. According to the meaning of the question, we get 2x + 3Y = 20 (and X and y are natural numbers) 2x = 20-3y, x = 10-32y  x = 10Y = 0, x = 7Y = 2, x = 4Y = 4, x = 1y = 6. So there are four ways to buy Coke and milk tea. The number of cups in each way is ① 10, 0; ② 7, 2; ③ 4, 4; ④ 1

Simplification (a + 1 / A-1) - (a square + A / a Square-1)

(a + 1 / A-1) - (a square + A / a Square-1)
=(a+1/a-1）-[a(a+1)/(a+1)(a-1)]
=(a+1/a-1)-a/(a-1)
=(a+1-a)/(a-1)
=1/(a-1)