What makes the second argument in Schaum's Outline Calculus more rigorous?

In summary, the conversation discusses the differentiation rule for the function ex and its inverse ln(x). It is shown that the derivative of ex is equal to ex, and a more rigorous argument is presented using Theorem 10.2(b). This theorem states that if a function is one-to-one, continuous, and has a non-zero derivative at a point, then its inverse is also differentiable at that point. The second argument provides a proof for this specific function.
  • #1
MaxManus
277
1
Hey, in my Schaum' Outline Calculus it says Dx(ex) = ex
Let y = ex. Then ln(y) = x. By implicit differentiation, [tex]\frac{1}{y}[/tex]y' = 1
therefor y' = y = exFor a more rigorous argument, let f(x) = ln(x) and f-1(y) = ey.
Note that f'(x) = [tex]\frac{1}{x}[/tex]. By Theorem 10.2(b).

(f-1)'(y) = [tex]\frac{1}{f'(f**-1(y))}[/tex],

That is Dy = [tex]\frac{1}{1/e**y}[/tex] = ey

10.2(b): Let f be one-to-one and continuous on the interval (a,b) Then:
If f'x(x0 is differentiable and f'(x0) != 0, then f-1 is
differentiable at y0 = f(x0) and (f-1)'(y0)
= [tex]\frac{1}{f'(x0}[/tex]

Now, my question is what does "more rigorous argument" mean and what makes the second one more rigorous.
 
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  • #2
The first argument uses the fact (your "10.2(b)") that if y= f(x) and x= f-1(y) then
[tex]\frac{dx}{dy}= \frac{1}{\frac{dy}{dx}}[/tex]
The second argument proves that is true for this particular function, as part of the proof.
 
  • #3
Thanks
 

1. What is a more rigorous argument?

A more rigorous argument refers to a logical and systematic approach to presenting evidence and reasoning to support a claim or hypothesis. It involves careful consideration of all relevant information and the use of established methods and principles to arrive at a sound conclusion.

2. Why is a more rigorous argument important in science?

In science, a more rigorous argument is crucial because it ensures that conclusions are based on reliable evidence and valid reasoning. This helps to minimize bias and errors, and allows for the replication and verification of results by other scientists.

3. What are the key components of a more rigorous argument?

The key components of a more rigorous argument include a clear and testable hypothesis, a thorough review of existing research, a well-designed experiment or study, accurate and precise data collection and analysis, and a logical interpretation of the results.

4. How does one develop a more rigorous argument?

Developing a more rigorous argument involves careful planning and execution of research, critical thinking and analysis of data, and constant evaluation and refinement of the argument. It also requires staying up-to-date with relevant scientific literature and seeking feedback from peers and experts in the field.

5. What are some common pitfalls to avoid in creating a more rigorous argument?

Some common pitfalls to avoid in creating a more rigorous argument include relying on personal beliefs or biases instead of evidence, using flawed or inadequate methods, and making unsubstantiated claims or generalizations. It is also important to acknowledge and address any limitations or potential alternative explanations for the results.

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