The differential of y = arctan (LNX)

The differential of y = arctan (LNX)

Guo Dunyong replied:
y = arc tan（ln x）,
y′=dy/dx=1/[ x（1+ ln²x）]=1/（x+ xln²x）,
dy=[1/（x+ xln²x）] dx.

Differential of x power of y = (LNX)
Differential of x power of y = (LNX)

Take logarithms on both sides and get the
lny=xln(lnx)
Derivation of X on both sides
1/y*y' = ln(ln(x)) + x (ln(ln(x)))'
= ln(ln(x)) + x (1/(xlnx))
=ln(ln(x))+1/lnx
And then we can substitute y and solve y '

How to use Cauchy's mean value theorem to find the limit of (x's square-x) / SiNx

Let f (x) = x ^ 2-x, G (x) = SiNx. Obviously, f (x), G (x) are differentiable on R and differentiable on [0, x]. If the condition of Cauchy mean value theorem is satisfied, then K belongs to [0, x]

4 (a-b) ^ 2 - (a + b) ^ 2

4(a-b)^2-(a+b)^2
=[2(a-b)-(a+b)][2(a-b)+(a+b)]
=(a-3b)(3a-b)

If there are five Saturdays and Sundays in March, what day is the first?

Friday

(2010 · Ji'an County) a yoghurt bottle (as shown in the picture), its body is cylindrical (excluding the bottleneck), and its volume is 32.4 cubic centimeters. When the bottle is placed upright, the height of the yoghurt in the bottle is 8 centimeters, and when the bottle is placed upside down, the height of the spare part is 2 centimeters. Please calculate, how many cubic centimeters is the volume of the yoghurt in the bottle?

8 + 2 = 10 (CM), 32.4 × 810 = 25.92 (cm3), a: the volume of yogurt in the bottle is 25.92 cm3

It is known that July 1, 2002 is Monday. What day is October 1, 2015?

There are 366 * 3 = 1098 days from 2004.2008.2012 to 2003, 2005.2006.2007.2009.2010.2011.2013.2014, 365 * 9 = 3285 days from July 1, 2002 to December 31, 2002, 31 + 31 + 30 + 31 + 31 = 184 days from January 1 to October 1, 2015

2 + 2 second power + 2 third power +... + 29th power + 2100th power
Let (1) x = + 2 + 2 & # 178; + 2 & # 179; + + 2 ^ 99 + 2 ^ 100, then
②2X=2＋2²＋2³＋2^4＋…… ＋2^100＋2^101,
Change the above equation: 2x = 2 + 2 & # 178; + 2 & # 179; + 2 ^ 4 + ＋2^100＋-2+2^101,
2x=x-2+2 ^101,
∴x=2 ^101-2
By using the above method, we can find out 1 / 2 + 1 / 2 + 1 / 2 + +One ninety ninth power of 2 + one hundredth power of 2

S=1/2+1/2²+1/2³+······+1/2⁹⁹+1/2¹⁰⁰ ①
2S=1+1/2+1/2²+1/2³+······+1/2⁹⁹ ②
② - ① s = 1-1 / 2 & # - 185; - # - 8304; - # - 8304;
That is: 1 / 2 + 1 / 2 & # 178; + 1 / 2 & # 179; + ······ + 1 / 2 & # 8313; - # 8313; + 1 / 2 & # 185; - # 8304; - # 8304; = 1-1 / 2 & # 185; - # 8304; - # 8304;

What is 42 times the unknown plus 25 times the unknown
Use equation, formulation

Let the unknown be x, 42x + 25X = (42 + 25) X
=67x

The shadow length of tree AB has been measured to be 9 meters. The sun is 30 degrees from the ground. The height of the tree is about 5.2 meters. Because of soil erosion, the tree falls down along the direction of sunlight, and the shadow length changes. Why is the maximum length of the shadow 10.4 meters?

The tree fell in the direction of the sunlight, and the sun fell in the direction of the ground