57. Find the differential of the following functions: (1)y=3x^2 (3)y=lnx^2 (5)y=e^(-x)*cosx (7)y=ln√(1-x^3) (9）y=tan(x/2)

57. Find the differential of the following functions: (1)y=3x^2 (3)y=lnx^2 (5)y=e^(-x)*cosx (7)y=ln√(1-x^3) (9）y=tan(x/2)

(1)y = 3x²
dy = (dy/dx)dx = 6xdx
[Note: this problem uses a simple power function derivation method]
(3)y = lnx^2 = 2×lnx
dy = (dy/dx)dx =(2/x)dx
[Note: this problem uses the derivative method of natural logarithm]
(5)y = e^(-x)*cosx
dy/ = (dy/dx)dx
= [-e^(-x)*cosx + e^(-x)*(-sinx)]dx
= -[e^(-x)]×(sinx + cosx)dx
[Note: this problem uses the derivative method of product]
(7)y = ln√(1-x³) = (½)ln(1-x³)
dy = (dy/dx)dx
= (½)[1/(1-x³)]×(-3x²)dx
= -3x²/[2(1-x³)]dx
[Note: this problem uses the derivative method of compound function]
(9）y = tan(x/2)
dy = (dy/dx)dx
= [1/cos²(x/2)]×(½)dx
= 1/[2cos²(x/2)]dx
Or = (& frac12;) sec & sup2; (x / 2) DX
[Note: this problem uses the derivative method of compound function]

Find the differential of the following functions
Find the differential of the following functions:
1．y=xarctan2x
2.y=ln(1+x^2)/(1-x^2)
3. Y = (under root (2-x ^ 2)) + xlnx

1.dy/dx=arctan2x + 2x/(1+4x^2)
2.y=ln(1+x^2) - ln(1-x^2)
So dy / DX = 2x / (1 + x ^ 2) + 2x / (1-x ^ 2)
3. Dy / DX = - X / (under root (2-x ^ 2)) + LNX + 1
Differential is very simple, just remember more formula, no skill

Sufficient conditions for differentiability of functions
The sufficient condition for the differentiability of function z = f (x, y) at point (x0, Y0) is that f (x, y) at point (x0, Y0) []
A. Two partial derivatives are continuous
B. The existence of two partial derivatives
C. There are directional derivatives in any direction
D. Functions are continuous and have partial derivatives

The continuity of function of two variables is a known condition. All you have to do is to prove that if the partial derivative is continuous, then the function of two variables is differentiable

The mighty monkey king can turn a somersault over 18000 miles, that is, 5.4 × 104KM, and Vega is about 24.5l. Y. (light years) away from the earth. How many somersaults does the monkey king have to turn to reach vega? If the monkey king turns a somersault every second, how many years does it take him to reach vega?

China's great mathematician Zu Chongzhi was the first to calculate the diameter of pi to several decimal places?

There's a dispute between six and seven, but the National Bureau of Education says seven. That should be seven

Calculate ∫ ∫ 3dydz + ydzdx + (Z ^ 2 + 2 * A / 3) DXDY, where the integral surface is the outer side surrounded by the cone x ^ 2 + y ^ 2 = (A-Z) ^ 2, z = 0, z = a

Let m be the integral of Z = 0
According to Gauss theorem
Original integral = ∫ ∫ (1 + 2Z) dv-m
=∫(1+2z)dz∫∫dxdy-(-∫∫(2a/3)dxdy)
=π∫(0->a) (1+2z)(a-z)^2dz +2πa^3/3
=(1/6)πa^3(a+2) +2πa^3/3
=(1/6)πa^3(a+6)

The circumference of a rectangle is 22.6 cm. After it is divided into two rectangles, the sum of the circumference is increased by 9 cm. The area of the original rectangle is
Next: what's the square centimeter?

（22.6-9）÷2×（9÷2）
=13.6÷2×4.5
=30.6

Find the upper and lower limit LIM (x approaches 0) ∫ (O-X) {under the root sign (1 + x ^ 2) DT} / X
It should be like this to find the upper and lower limit LIM (x approaches 0) {(1 + x ^ 2) d} / x under the root of ∫ (O-X)

X approaches to 0, ∫ (0-x) {1 + T ^ 2) DT} approaches to 0, using Robida's rule: LIM (x approaches 0) ∫ (0-x) {1 + T ^ 2) DT} / x = LIM (x approaches 0) d / DX ∫ (0-x) {1 + T ^ 2) DT} = LIM (x approaches 0) DT (1 + x ^ 2) = 1

The known equation x2 / (M - 3) + Y2 / (5-m) = 1 represents an ellipse with focus on the y-axis

X2 / (M - 3) + Y2 / (5-m) = 1 represents the ellipse with focus on the Y axis
Then 5-m > imi-3
ImI+m

The ratio of the smallest prime to 6 and the ratio of 1 / 2 to X are equal

Minimum prime = 2
2：6=1/2：x
2x=6*1/2
2x=3
x=3/2