Let y = e ^ xsinx, find y '

Let y = e ^ xsinx, find y '

y'=e^x *sinx+e^xcosx=e^x*( sinx+cosx)

The differential of y = xsinx / 1 + TaNx function

y=xsinx/(1+tanx)
=x/(1/sinx+1/cosx)
=x/(secx+cscx)
(secx)'=tanx·secx
(cscx)'=-cotx·cscx
y' = [(secx+cscx)-x(tanx·secx-cotxcscx)] / (secx+cscx)^2
So: dy = {[(secx + CSCX) - x (TaNx · secx cotxcscx)] / (secx + CSCX) ^ 2} DX

In all permutations A1, A2, A3, A4, A5 of 1, 2, 3, 4, 5, (1) find the probability of satisfying A1 & lt; A2, A2 & gt; A3, A3 & lt; A4, A4 & gt; A5; (2) note that ξ is the number of permutations satisfying AI = I (I = 1, 2, 3, 4, 5), find the distribution sequence and mathematical expectation of ξ

(1) In the permutation satisfying A1 & lt; A2, A2 & gt; A3, A3 & lt; A4, A4 & gt; A5, if A1, A3, A5 take the elements in the set {1, 2, 3} and A2, A4 take the elements in the set {4, 5}, all of them

The area of a trapezoid is 18 square meters, its upper bottom is 6 decimeters, and its height is 3 decimeters. How many decimeters is its lower bottom? Solve the equation

Set the height to h decimeter
（6+H）×3÷2=18
6+H=18×2÷3
H=12-6
H=6

If f (2x + 1) = lgx, then f (21)=______ ．

According to the meaning of the question, let 2x + 1 = t (t ＞ 1), then x = 2T − 1, that is, f (x) = lg2x − 1 (x ＞ 1); {f (21) = lg221 − 1 = - 1

It is known that the length of the classroom is 9 meters, the width is 7 meters, the height is 3 meters, and the area of the doors and windows is 18.5 square meters
If it costs 6 yuan per square meter, how much does it cost to paint the classroom? Please write the specific steps

(9 * 3 + 7 * 3) * 2 + 9 * 7-18.5 = 140.5 square meters * 6 = 843 yuan, appraisal completed! Shanghai Gulan construction waterproof Engineering Co., Ltd. also undertakes all kinds of interior and exterior wall painting projects!

Hello, there's another problem. If you pass through the chord ab of a point m (1,1) in the ellipse x ^ 2 / 16 + y ^ 2 / 4 = 1, you can find the trajectory equation of the midpoint of the chord passing through the point M,

Let a (x1, Y1), B (X2, Y2), and P (x, y) be the midpoint, then X1 + x2 = 2x, Y1 + y2 = 2Y. On the one hand, substituting a and B into the elliptic equation, we get X1 & # 178; + 4Y1 & # 178; = 16 (1) x2 & # 178; + 4y2 & # 178; = 16 (2) (2) - (1) to get (x2-x1) (x1 + x2) + 4 (y2-y1) (Y1 + Y2) = 0. When x1 ≠ X2, k = (y2-y1) / (x2

The area of a triangle is half that of a parallelogram

The area of a triangle is half that of a parallelogram
Should be:
The area of a triangle is half that of a parallelogram with the same base and height

Given the fixed point a (1,3), the moving point P moves on the ellipse x ^ 2 / 4 + y ^ 2 = 1, and the other moving point m satisfies the vector am = 2 and the vector MP, the trajectory equation of the moving point m is obtained

Let m (x, y), P (x0, Y0)
Because am = 2MP, a (1,3)
So (x-1, Y-3) = 2 (x0-x, y0-y)
That is, X-1 = 2 (x0-x), Y-3 = 2 (y0-y)
So x0 = (3x-1) / 2, Y0 = (3y-3) / 2
Because P (x0, Y0) is on the ellipse x ^ 2 / 4 + y ^ 2 = 1
So [(3x-1) / 2] ^ 2 / 4 + [(3y-3) / 2] ^ 2 = 1
That is, (3x-1) ^ 2 / 16 + (3y-3) ^ 2 / 4 = 1

For any positive integer n, it is proved that the value of (3N + 1) (3n-1) - (3-N) (3 + n) is a multiple of 10

It is proved that: the original formula = (3N + 1) (3n-1) - (3-N) (3 + n) = 9n2-1 - (9-n2) = 10n2-10 = 10 (n + 1) (n-1), ∵ n is a positive integer, and (n-1) (n + 1) is an integer, that is, the value of (3N + 1) (3n-1) - (3-N) (3 + n) is a multiple of 10