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The Derivative of the Exponential Function, base e:

Updated on April 4, 2013

1. Paragraphs 5, 6, and 7 address heaven or hell--a choice to be made until death.


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?


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?


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.


E4.1 Catman has several examples( linked) of how to differentiate the base e to a functional power( ef( x) ).


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.


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.


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).


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    • Caleb DRC profile image

      Caleb DRC 5 years ago

      Hi North Wind, my mouser is clicking on his own, so I end up having to delete duplicate replies.

      I'm a firm believer in Romans 8:38,39; however, I think the Scriptural evidences is that I can separate me from the love of God. I can constantly choose to walk slowly or run quickly from Christ, and eventually it may be too late to recover me. I believe God does not let us go without a fight as with chastisement and continued warnings, but there is a reason we must work out our own salvation in fear and trembling( Philippians 2:12).

      You're right, "There is, however, no proof of that in the Bible." This is typical of the Catholic church: believing in many things that can not be backed up with Scripture, and believing in doctrines that Scripture commands us to not believe in or adhere to.

    • Caleb DRC profile image

      Caleb DRC 5 years ago

      Hyphenbird, I think you hit it on the nail--right on target. It is mandatory and central to what you said, "Decide to believe in and obey Christ."( Joshua 25:15, with verse 14). We must decide and then we must choose, and then stick to that choice. God has inundated us with evidence of His existence in both omniscience and omnipotence in what we have seen, and continue to discover in creation. Jesus Christ has also inundated us with evidence that He is the Son of God. No other "prophet" from "other religions" has( or had) His credentials as healing the sick, changing the weather, walking on water and raising the dead. Therefore, God has never expected or demanded blind faith from us. So it is sad indeed that people do not give Him a chance and choose life.

    • Hyphenbird profile image

      Brenda Barnes 5 years ago from America-Broken But Still Beautiful

      Decide to believe in and obey Christ. Amen. He certainly made me a new woman and I truly understand the concept of "born again". I wish everyone would give Him a chance and choose life.

    • North Wind profile image

      North Wind 5 years ago from The World (for now)

      Hi Caleb DRC,

      Your Math is amazing as usual but as usual I am drawn like a firefly is to a light to the topic of God and salvation. The debate about salvation and whether you can lose it or not is a big one. I do believe that He finishes the work He starts in you but the evidence of His work in you is your glorification of Him. As He said, we will know them by their fruit.

      It is interesting that you brought up the Roman Catholic faith who do believe that there is hope after the grave and often pray for the dead. I think that there are many who think that perhaps, God will give them a chance after death and that some hold to Roman Catholic doctrines without even knowing that they are R.C. doctrines. There is, however, no proof of that in the Bible.

      Thank you for your sober warning. I hope that many will come by and read this. Who knows when our appointments with death may come?