Different proof of the derivative of e^x

In summary: So in general, this statement is not true. In summary, the conversation discusses a proof of the derivative of e^x and the use of infinitives in mathematical calculations. The person providing the summary explains that the original proof is not valid due to errors such as using the same symbol for two different variables and plugging in variables that are already present in the expression. They also clarify that the statement "1 = \lim_{y \rightarrow 0} m(y)y" is not true in general, unless m and y are dependent on each other.
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  • #2
On the third-to-last line you could have just subtracted out the n terms. This would allow you to not make the mistake of assuming infinity - infinity = 0.
 
  • #3
No, that's not a valid proof. You've made several errors. For example, the claim

[tex]\lim_{y\rightarrow 0} \frac{z}{y} = \lim_{n\rightarrow \infty} zn[/tex]
is valid due to the change of variables 1/y = n, but you then go and plug this into a formula which already has a variable labelled n which is different and indepedent from1/y. So, you should more appropriately have labelled it 1/y = m, so m and n remain independent. Your entire result relies on using the same symbol for two different variables.

Another error is that you cannot multiply "rule B" by y. Given 1/y = m you can conclude 1 = my, but you cannot make the claim that
[tex]\lim_{y \rightarrow 0} 1/y = \lim_{m\rightarrow \infty} m \Rightarrow 1 = \lim_{y\rightarrow 0}\lim_{m \rightarrow \infty} my[/tex]

The implication is not correct.
 
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  • #4
I guess my original question was are these assertions correct independently, even if i didn't prove them?

Is this true?
[tex]\lim_{a\rightarrow 0} \frac{b}{a} = \lim_{d\rightarrow \infty} bd[/tex]

[tex] 1 = \lim_{y\rightarrow 0}\lim_{m \rightarrow \infty} my[/tex]
 
  • #5
Natel said:
I guess my original question was are these assertions correct independently, even if i didn't prove them?

Is this true?
[tex]\lim_{a\rightarrow 0} \frac{b}{a} = \lim_{d\rightarrow \infty} bd[/tex]

If you define 1/a = d, then yes, this is true. However, if you plug this into an expression which has a variable already labelled a or d, then you have to use a different letter to denote one of these variables, because they are not the same. This is a mistake you made in your proof: you set 1/y = n, but you already had an n in your expression, so you should have let 1/y = m. Your proof would have then had to work out different, because your manipulations depended on the confusion between the n from "1/y = n" and the n that was already present in your expression.

[tex] 1 = \lim_{y\rightarrow 0}\lim_{m \rightarrow \infty} my[/tex]

If m and y are independent, no. The right hand side is an indeterminate expression. If m depends on y somehow, then it could be true, but then you would only have one limit on the right hand side because m is a function of y. If m = 1/y, then you get
[tex]\lim_{y \rightarrow 0} m(y)y = y/y = 1,[/tex]
but if m = ln(y), you get
[tex]\lim_{y \rightarrow 0} m(y)y = y\ln(y) = 0.[/tex]
 

1. What is the derivative of e^x?

The derivative of e^x is also e^x. In other words, the derivative of e^x is itself.

2. Can you prove the derivative of e^x using the limit definition?

Yes, the derivative of e^x can be proven using the limit definition. By taking the limit as h approaches 0 of (e^(x+h) - e^x)/h, we can show that the derivative of e^x is e^x.

3. Is there another way to prove the derivative of e^x?

Yes, there are several other ways to prove the derivative of e^x. One method involves using the chain rule and the fact that the derivative of e^x is e^x. Another method involves using the power rule and the fact that e^x can be rewritten as (e^1)^x.

4. Why is the derivative of e^x important?

The derivative of e^x is important because it allows us to find the rate of change of exponential functions. It is also a fundamental concept in calculus and is used in many applications, such as in modeling growth and decay.

5. Can the derivative of e^x be negative?

Yes, the derivative of e^x can be negative. The sign of the derivative depends on the value of x. For example, when x is negative, the derivative of e^x will also be negative.

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