Derivative of Arcsin X and Arcsin X 2

Derivative of Arcsin X
If there is an arcsin function X, then the derivative of arcsin is given by dividing one by the square root of the value obtained from subtracting the square of x from one. This is shown as a formula below:

Derivative of ArcSin = 1/square root of (1-square of X)

Here X belongs to real number in such a way that its value lies between -1 and +1.

Let us prove the formula for derivative of arc-sin X.  If Y = arcsin X where X is a real number lying between -1 and +1, then we can say that X = sin Y. By using the derivative of sine function, we can rewrite in the derivative form as dX/dY = cos Y. By applying the derivative of inverse function, it becomes dY/dX = 1/cos Y. We know that cos square Y + sin square Y is equal to 1. We can rewrite it as cos Y equal to plus or minus of the square root of one minus sin square Y. As cos Y is greater than or equal to Zero on the arcsin X range that ranges from –pi/2 to +pi/2, we can say that cos Y always takes a positive value. So we get the value of dy/dx as one divided by the square root of the value obtained from subtracting sin square y from one.  In other words, the differentiation or the derivative of arc-sin X is equal to one divided by the square root of the value obtained from subtracting the square of X from one, as we know X = sin Y. Thus, Derivative of Arc-Sin = 1/square root of (1-square of X)

Derivative of Arcsin X 2
As per the chain rule, the derivative of a square of any function is given by twice the function itself. By applying this rule, we can find out the derivative of arc-sin X2 from the formula of       arcsin X.

We have seen that g(X) = arcsin X. We know the differentiation of arcsin X i.e. d/dx arcsin(X) is obtained from the formula 1/square root of (1-square of X). By applying the chain rule, we get 2arcsin(X) multiplied with (1/square root of (1-square of X)). Thus, Derivative of arc-sin X2 = 2arcsin(X)/square root (1-square of X).

Derivative of arcsin(X)+arcos(X)
The differentiation of the summation of arcsin(y) and the arcos(y) is given by pi divided by 2, i.e. arcsin(X) +arccos(X) = pi/2.

Applications of Arcsin
An important application of the arcsin transformation can be seen in the analysis of binary data.

Measures of variation

Measures of variation definition that is the difference between highest value and the lowest value. That is a particular range.  In another way we also simplify that Measures of variation means the quantities that show the measure of variation in terms of random variable. For better understand the measure of variation we should have knowledge of random variables.

Random variable is determined by the outcome of the experiment which we have done. We may assign probabilities to the possible values of the random variable. We can also differ random variable in two ways. First one is discrete random variable in which a variable that can take one value from a discrete set of values. And the second one is continuous random variable, which means a variable that can take one value from a continuous range of values.

Measures of variation in statistics, statistics is a branch of mathematics which gives us the tools to deal with large quantities of data and derive meaningful conclusions about the data. To do this, statistics uses some numbers or a measure which describes the general feature contained in the data. In other word using statistics we can summaries large quantities of data by a few descriptive measures.  In this section we describe how any value is carried out from a given set of data. Measures of the variations sometimes are also known as measure of dispersion or measure of spread.

In statistics there are three measures of the variation…

1. The range
2. The standard deviation
3. The variance

The standard deviation is a measures of the variation or measure of dispersion amongst data. Instead of taking absolute deviation from the arithmetic mean we may square each deviation and obtain the arithmetic mean of squared deviations. This gives us the variance of the values. The positive square root of the variance is called the standard deviation of the given values. The standard deviation may be for raw data and grouped data.

The variance means the square of standard deviation. Alternatively square root of variance is equal to standard deviation. The larger the standard deviation, larger will be the variance. The standard deviation is an absolute measure of variation and hence cannot used for comparing variability of two data sets with different means. Therefore such comparisons are done by using a relative measure of variation known as coefficient of variance.

At last we must take measure of variation examples. Suppose, we have a set in which five different numbers so to find measure of variation we have to calculate only range means difference between highest and lowest value in given set of value.

One Sided Limits: The Right and Left hand limits

One Sided Limits

Limit of a function is given by lim (x->a) f(x) = L, that is, as x approaches ‘a’ from the right hand side of a, f(x) = L and also, as x approaches a from the left hand side of a, f(x)=L.

One Sided Limits Examples
Let us consider a function f(x) = -x^2+6 for  0<=x<2 and f(x)=x-1 for  2<= x <= 4. When we graph this function, we can see that the function is discontinuous at x=2, and the two pieces of the function approach different values, we get, lim (x->2)f(x) is undefined. We can say that the limit of f(x) as x approaches 2 from the left is 2, and the limit of f(x) as x approaches from the right is 1. We can write it as, lim (x->2-) f(x) = 2 and lim (x->2+) f(x) = 1. The plus sign indicates ‘from the right side’ and the minus sign indicates ‘from the left side’.

There are two one sided limits, Right and left hand limits. Right hand limit which is defined as lim (x->a+) f(x) = L , if f(x) is as close to L as we want for all x sufficiently closer to a and x is greater than a, without actually letting x equal to a. Left Hand Limit is defined as lim(x->a-)=L, if f(x) is as close to L as we want for all x sufficiently closer to a and x is less than a, without actually letting x equal to a.

Finding One Sided Limits Algebraically
Let us consider a piecewise function f(x)= x+2 for x less than or equal to 6 and f(x) = x^2-1 for x greater than 6. The steps involved in finding one sided limits of the given piecewise function are as follows:

Left Handed limit of f(x) is given by lim (x->6-)f(x), all the values of x on the number line lie to the left of 6 which means  x is less than 6 and hence we consider the function f(x) = x+2. We plug in the value x=6 in the function which gives, 6+2 = 8

Right Handed Limit of f(x) is given by lim(x->6+) f(x),  all the x values considered on the number line lie to the right of 6, so, x is greater than 6 and hence we consider the function f(x) = x^2-1. We shall plug in the value x=6 in the function which gives, (6)^2-1 = 36-1=35
Lim(x->6)f(x) does not exist as the right handed limit is not equal to the left handed limit

Continuous Variable

Variable is just like a box where we can store the property of an object. The property of an object may be a value or a characteristic, variable takes more than one value. For example, “marks” of a student in an exam is a variable. Eye color of individuals is a variable which is measured on nominal scale. Height of individuals is also a variable which is measured on ratio scale. Variables are classified into two categories those are Continuous Variable and Discrete Variable.

Continuous Variable Definition: A variable is said to be continuous variable if takes each and every value in between two values (or limits). Continuous variable Example is given below.

• “Rain fall in a day” is a continuous variable it can take any value in between 0 and infinity.
• “Time to complete this answer” is a continuous variable it can take any value between 0 and infinity.
• “Water level in a tank” is a continuous variable it can take any value between 0 and maximum capacity of tank.

Simultaneous Equations

Equations
Any expression F(X) when equated to an expression or a constant is called an equation.
When the highest power of X is one it is called a simultaneous equation.

What are Simultaneous Equations
A set of equations having many variables, is called simultaneous equations. To solve a simultaneous equation of two variables we need minimum two equations, which are different from each other.

Simultaneous Equation

Ex: 2x+3y = 5 and 4x +3y = 10.

Solving Simultaneous Equations:

Substitution method
Algorithm:
Step 1: Express one of the variables in terms of the other.
Step 2: Substitute the value of the variable in the second equation to obtain an equation in terms of one single variable.
Step 3: Solve for the variable.
Step 4: Substitute the value of the variable in the expression to get the solution of the other variable.

Solved Example:
Solve 2x +3y =5; 5x +2y=7
Step 1: 2x= 5-3y; x= (5-3y)/2
Step 2: Substituting the value of x in the second equation we get
 5((5-3y)/2) + 2y=7 Solving for y we get
 (25-15y+4y)/2=7
 (25-11y)/2=7
 25-11y=14
 25-14=11y
 11=11y
 y=1
Step 3: Substituting the value of x in step 1 we get x=(5-3(1))/2 =1
Hence the solution set is X=1, y=1 or {1, 1}

Method 2:
Elimination Method:
Algorithm:
Step 1: Make the coefficients of one of the variables same.
Step 2: If the signs are same, subtract and if the signs are different add them to obtain a linear equation of one variable and solve for it.
Step 3: Substitute the obtained value in any one of the equations to solve for the other variable.
Solved Example:
4x+ 3y = 11; 3x +2y=8
4x+3y=11  --------> multiply 2
3x+2y=8  --------> multiply 3
8x + 6y =22
9x + 6y =24 (as the signs are same subt)
(-)   (-)     (-)
-x = -2 or x=2
Substituting the value of x in the first equation we get
4(2) +3y =11;
8+3y=11;
3y= 11-8;
3y=3; y=1.
The solution set is x= 2 and y = 1.
Cross Multiplication Method
Algorithm:
Step 1: Rearrange the equations as shown.
If the equations are as follows

Step 2: Cross-multiply and subtract starting from top end corner of the denominator and write as follows

Step 3: 
Equate the x term with constant term and solve for x. Similarly equate y term with constant term and solve for y.
Solved Example:
2x +3y = 7
6x -5y =11

Cross-multiplying and subtracting we get

Product Rule for Derivatives

Product Rule for Derivatives:

The Product Rule helps us to differentiate a function when the function is a product of two distinct functions.

 Product Rule is necessary whenever we need to differentiate a function which is a product of two sub-functions that are easier to differentiate.

Let us consider some examples for a better understanding of the Product Rule for Derivatives

Associative Property

Definition of associative property:

Associative property pertains to number, sets and many other mathematical components. The word associative is derived from the word ‘association’, which means being related to. In math, we define associative property for various operations on numbers or sets etc.

Associative Property

What is associative propery?

All types of numbers, whether natural, whole, rational, irrational and real numbers have associative properties. There are mainly two types of associative property for numbers. They are:

1. Associative property of addition:
Addition associative property definition is as follows: Consider three numbers a, b and c. then, (a+b) + c = a + (b+c). Meaning we associate a with b first then associate the sum with c or we associate a with sum of b and c, either ways, the result would be the same.
For example: (1/2 + 1/3) + ¼ = ½ + (1/3 + ¼) = 13/12

2. Associative property of multiplication:
Numbers obey associative property in multiplication as well. Consider three numbers, p,q and r, then (p x q) x r = p x (q x r). That means whether we take product of p and q and then multiply with r or we multiply p with product of q and r, either ways we would get the same answer.

For example: (1/2 x 1/3) x ¼ = ½ x (1/3 x ¼) = 1/24
Associative property with respect to sets:

Consider three non disjoint sets, A, B and C. Then both the operations of union and intersection on these sets is associative. That means, A U (B U C) = (A U B) U C and
A ∩ (B ∩ C) = (A ∩ B) ∩ C.

Example: Consider the following sets: A = {1,2,3,4,5}, B = {2,4,6,8,10}, C = {3,4,5,6,7}
Then, A U (B U C) = {1,2,3,4,5} U ({2,4,6,8,10} U {3,4,5,6,7}) = {1,2,3,4,5,6,7,8,10}
And (A U B) U C = ({1,2,3,4,5} U {2,4,6,8,10}) U {3,4,5,6,7} = {1,2,3,4,5,6,7,8,10}
So we see that both are equal. Similarly .
A ∩ (B ∩ C) = {1,2,3,4,5} ∩  ({2,4,6,8,10} ∩  {3,4,5,6,7}) = {4,6} and
(A ∩ B) ∩ C = ({1,2,3,4,5} ∩  {2,4,6,8,10}) ∩  {3,4,5,6,7} = {4,6}