next we will be searched because  then with L' Hospital rule 2 let  let differentiable at  Such that  for all . suppose that   If , then If  , then so that,  then so it is proven that

Examples: Find the derivative of x from the following functions y = xx y = sin(3x+2) y = cx y = cos(4x) Answer: y = xx xx  = exlnx g = ex dan f = xlnx then (g o f)(x) = exlnx = xx so that, (g o f)’(x) = g’(f(x))f’(x) = (exlnx)(d(xlnx)/d(x)) = (xx)((1.lnx)+(x.(1/x)) … Lanjutkan membaca Solutin with Chain Rule on Real Analysis

Definition 1 (homomorphism Semigroup) Let (S, ∗) and (T, ⋆) be semigroup. Mapping f: S ⟶ T is called a homomorphism Semigroup if for every x, y in S, satisfies f (x ∗ y) = f (x) ⋆ f (y) Example 1: The set of all enumeration numbers ℕ + = {0, 1, 2, 3, … Lanjutkan membaca Homomorphism Semigroup

3.1.8 definition If X = (x1, x2, .., xn, ...) is a sequence of real numbers and if given m natural numbers, then m-tail of X is a sequence. Xm = (xm + n: n ϵ N) = (xm + 1, xm + 2, ...) Example: 3-tail of sequence X = (2, 4, 6, 8, … Lanjutkan membaca Real Analysis : Tails of Sequence

3.1.3 Definition Sequnce X = (xn) on R are said to converge to x ϵ R, or x is said to be the limit of (xn), if every 𝛆> 0 there is a natural number K (𝛆) as for all n ≥ K (𝛆), the limit xn meets | xn - x | between xn … Lanjutkan membaca Real Analysis : Sequence and Their Limits

Definition: Each set of restricted real numbers has a supremum. If a set A is limited to bottom then a -A assembly is limited to the top. According to the above definition, -A soup exists. As a result inf A = - A -sup. This means that a limited set is down to a minimum. … Lanjutkan membaca Real Analysis : Complementarity of Real Numbers and Archimedes Properties

given a non empty set A subset of R, The number x is called the upper bound of the set A if for every aЄA a is applied. the number A is said to be restricted to the top if it has an upper bound. the number x is said to be no upper bound … Lanjutkan membaca Real Analysis : Supremum and Infimum

RING FACTORS (RING QUOSIEN) Based on Group Theory, if the H subgroup of (R, +) and at R is subject to the relation "a ~ b" if and only if a-b∈H. for every a, b∈R, the relation ~ is an equivalent relation. Therefore, R is split in equivalence classes. The set of all equivalence classes … Lanjutkan membaca Ring Factors (Ring Quosien)

Definitions 1: A binary relation on set is called a partial order / partial ordering on if it is: 1. REFLECTION: if the relation R on A is called reflexive properties, if every element of A associated with him. For each a∈A, (a, a) ∈A or For each a ∈A, aRa 2. ANTISIMETRY: for each … Lanjutkan membaca Partial Order Relation