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The Derivative of the Exponential Function, base e:
1. Paragraphs 5, 6, and 7 address heaven or hell--a choice to be made until death.
THE STRUCTURE OF SPACE AND e:
2. Just what is the structure of space? Our scientists and mathematicians have proven that space is real; i.e. it has physical essence and structure. Hub #14.1( link to it) mentions at paragraph 9 that information may be stored in its structure and at par. 10 it says space can be both stretched and spread out. Paragraph 21 of hub#12.5(linked) explains further how information can be stored in space. So how does "e" fit into all of this? Par. 3 of hub#12.14( linked) mentions how e is associated with many aspects of nature as growth and decay rates, chemical processes, thermodynamic processes, and it is ubiquitous( par. 2 of hub#12.14) in the mathematics that describes those rates and processes. The constant e was described as a limit at hub#12.14, but it can also be defined as an infinite series as shown at L1 below. This constant, e, is involved in all of these things, so how does its infinite series fit into all of this? What physical phenomenon or structure does it describe, in any? Currently I think only God has the answer to that question. It may give us some insight into the structure of space itself, or to the forces, as the weak nuclear force, for example. Perhaps it gives information to the permeability or permittivity of a vacuum. I do think the infinite series that represents e is giving us significant insight into something . . . but what?
L1 THROUGH L8 EXPLAINED:
3. L1 through L2.2 shows how that particular infinite series generates the constant e. You can do the arithmetic on the series as long as you want, to get whatever accuracy you want for e. L3 is the infinite series that generates e raised to the X( variable) power. e^7 equals 1096.633158. If you plug 7 in for X in the series of L3 and take it out to 20 terms you will get an answer close to what was just given( 1096.633158). The derivative of e^X is just e^X, which is another remarkable characteristic of this constant. This is proven at L5 through L8. If we differentiate each term of the infinite series( at L3) that represents e^X, then we end up with the same series at L8. This was done from L5 through L8. Amazing, isn't it?
L9 THROUGH L13.1 EXPLAINED:
4. L9 through L11 is another way to prove that the derivative of e^X is just e^X. Here implicit differentiation is being used. The reason why the chain rule is used at L10 is because Y is a function of X. It is not X, but rather is a type of composite function. I do not think it would be appropriate to call Y a composite function, but I think of it that way so I'm reminded that the chain rule is to be used in implicit differentiation. At L10 we multiply both sides of the equation by Y, and get dy/dx = Y, but Y = e^x; therefore, the derivative of e^x is just e^x, which is shown at L11. From L12 through L13.1 I give another example of how the chain rule is used. Let us say that U at L12 is a function of X. Then when finding the derivative, we differentiate e^u but then multiply this with the derivative of U. I give an example of this at L13 through L13.1. The commutative property of multiplication was used to bring cosX from the right to the left at L13.1.
A LINK FOR MORE EXAMPLES:
E4.1 Catman has several examples( linked) of how to differentiate the base e to a functional power( ef( x) ).
CAN WE LOOSE OUR SALVATION?
5. A verse used to prove we can not loose our salvation is Romans 9:39, but it says at the end, "which is in Christ Jesus our Lord." We must remain in Christ to keep our salvation. Remaining in Christ is obeying Christ; this is saving faith. We can't call Christ Lord if we don't do what He said to do( Luke 6:46). Only then do we have stability of salvation( Luke 6:47-49). Mercy is extended to those who obey( Exodus 20:6; Deuteronomy 5:10; Daniel 9:4; John 14:15, 23; 15:10). Yes, Christians can and do sin, which John does acknowledge( 1 John 2:1), but right after acknowledging that he says a person does not even know God if he sins( Verse 4). This is not a contradiction. It is stating two facts: 1. Christians sin, 2. It is imperative they stop. Remember the frog at paragraph 12 of hub#12.12( linked). Many Christians believe this is not possible, but it is, which is why God has so many warnings about it.
THE EVIDENCE IMPLIES WE CAN LOSE OUR SALVATION:
6. One is not in the Book of Life unless he is saved, but God clearly states he/she can be blotted out( Exodus 32:33; Revelation 3:5). God did not keep the angels who sinned against Him( 2 Peter 2:4; Jude 1:6), or His own people( Romans 11:20, 21; Jude 1:5). Also Ezekiel 3:20; 33:12, 13, 18 implies the righteous can lose their salvation if they choose to sin, and Hebrews 6:4-6 gives one of the most horrible warnings concerning this subject.
NO SECOND CHANCES AFTER DEATH:
7. Yes, we get second chances and more in this life, but not after we die--contrary to what the Catholic church may teach. This is it folks; this is our probationary time. If we mess this up and die, then it is over--no more chances. We are talking eternal life, or eternal torment. Do not choose eternal torment. Choose eternal life! Decide to obey Christ( Joshua 24:14, 15).