Is the Derivative of an Inverse Function Valid? Insights and Links!

In summary, the conversation discusses the validity of a proof for the derivative of an inverse function using the definition of a derivative and the chain rule. There is some confusion about the use of x and y in the proof, but overall it is considered valid. The conversation also briefly mentions the use of rational and real numbers in the proof.
  • #1
Petrus
702
0
Hello MHB,
I am aware of there is two way, u can use chain rule or defination of derivate. I totaly understand the proof with this type Derivative of Inverse Function but is that a valid proof? How ever our teacher did proof this with derivate defination which I don't understand from my textbook. What is your thought? Any good link that explain this proof with derivate defination

I am aware that we use chain rule and I am training for oral exam and I guess I will have to proof this chain rule in this one.

edit: why should \(\displaystyle f'(x) \neq 0\) should it be \(\displaystyle f'(y) \neq 0\)
Regards,
\(\displaystyle |\pi\rangle\)
 
Last edited:
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  • #2
re: proof of inverse derivative

Petrus said:
Hello MHB,
I am aware of there is two way, u can use chain rule or defination of derivate. I totaly understand the proof with this type Derivative of Inverse Function but is that a valid proof? How ever our teacher did proof this with derivate defination which I don't understand from my textbook. What is your thought? Any good link that explain this proof with derivate defination

I am aware that we use chain rule and I am training for oral exam and I guess I will have to proof this chain rule in this one.

edit: why should \(\displaystyle f'(x) \neq 0\) should it be \(\displaystyle f'(y) \neq 0\)
Regards,
\(\displaystyle |\pi\rangle\)

That proof looks valid to me.

Note that there may be some confusion about x and y, since their meanings are swapped around after the first line.
In the first line x is used as the argument of f, but in the second line and thereafter x is used as the argument of $f^{-1}$ instead (where you might expect y to be the argument).
 
  • #3
Re: proof of inverse derivative

I like Serena said:
That proof looks valid to me.

Note that there may be some confusion about x and y, since their meanings are swapped around after the first line.
In the first line x is used as the argument of f, but in the second line and thereafter x is used as the argument of $f^{-1}$ instead (where you might expect y to be the argument).
Thanks for taking your time I like Serena!:)

PS. Should I be rational or real:p

Regards,
\(\displaystyle |\pi\rangle\)
 
  • #4
Re: proof of inverse derivative

Petrus said:
Thanks for taking your time I like Serena!:)

PS. Should I be rational or real:p

Regards,
\(\displaystyle |\pi\rangle\)

I think that \(\displaystyle |\pi\rangle\) is imaginary. (Pizza)
 

Related to Is the Derivative of an Inverse Function Valid? Insights and Links!

1. What is "Proof of inverse derivative"?

Proof of inverse derivative is a mathematical concept used to prove that the inverse of a differentiable function is also differentiable. It is a way to show that the inverse function has a well-defined derivative at a given point.

2. Why is "Proof of inverse derivative" important?

"Proof of inverse derivative" is important because it allows us to determine whether the inverse of a function is also differentiable. This is useful in many applications, such as optimization and curve fitting.

3. How is "Proof of inverse derivative" used in real-world situations?

"Proof of inverse derivative" is used in real-world situations in which we need to find the optimal value of a function. For example, in economics, we may need to find the price that maximizes profit, and "proof of inverse derivative" helps us find this value.

4. What are some common techniques used in "Proof of inverse derivative"?

Some common techniques used in "Proof of inverse derivative" include the chain rule, implicit differentiation, and the definition of the derivative. These techniques allow us to manipulate the function and its inverse to show that the inverse is also differentiable.

5. Are there any limitations to "Proof of inverse derivative"?

Yes, there are limitations to "Proof of inverse derivative." One limitation is that it only applies to functions that are differentiable. If a function is not differentiable, then its inverse may not be differentiable either. Additionally, "proof of inverse derivative" may not work for functions with sharp corners or discontinuities.

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