How Do I Apply L'Hopital's Rule to Exponential Derivatives?

In summary, the problem is that you are unfamiliar with the L'Hopital Rule for derivatives, and you are looking for the derivative of a function that does not have a derivative.
  • #1
KristinaMr
11
1
Homework Statement
how can I find the derivative of (2x-1)^1/(x-1)...the second part is all in the exponent
Relevant Equations
I really don't know which rule applies in this case..maybe there's a way to rearrange the expression?..any help is appreciated
I encountered this problem is Hopital section...how do I even apply it?
 
Physics news on Phys.org
  • #2
[tex]
(2x - 1)^{1/(x-1)} = \exp\left(\frac{\ln (2x-1)}{x - 1}\right)
[/tex]
 
  • Like
Likes KristinaMr and FactChecker
  • #3
Do you mean L'Hopital? I don't know how that applies to this derivative either. Regarding the derivative, are you familiar with the Exponent rule for derivatives? (see Exponent Rule for Derivatives )
 
  • #4
FactChecker said:
Do you mean L'Hopital? I don't know how that applies to this derivative either. Regarding the derivative, are you familiar with the Exponent rule for derivatives? (see Exponent Rule for Derivatives )
thank you for the exponent rule ..I somehow missed it

yea it was a limit of x-> 1 so that the exponent would be 1/0 ..the exercise said to apply L Hopital rule to solve ( by the way the result should be e)
 
  • #5
KristinaMr said:
Problem Statement: how can I find the derivative of (2x-1)^1/(x-1)...the second part is all in the exponent
Relevant Equations: I really don't know which rule applies in this case..maybe there's a way to rearrange the expression?..any help is appreciated

I encountered this problem is Hopital section...how do I even apply it?
This problem has nothing to do with finding the derivative of that function.

Write the function above as an equation: ##y = (2x - 1)^{1/(x - 1)}##
Take the log (ln) of both sides to get ##\ln y = \frac 1 {x - 1} \ln(2x - 1) = \frac{\ln(2x - 1)}{x - 1}##
Now take the limit as x --> 1 of both sides. At this point the limit on the right side can be evaluated using L'Hopital's Rule.

Your textbook should have and example of this technique. Look at the example, and follow the steps in that example.

KristinaMr said:
by the way the result should be e
I don't think so, not as you have shown the problem. I get a value of ##e^2## for the limit.
 
  • #6
Mark44 said:
I don't think so, not as you have shown the problem. I get a value of ##e^2## for the limit.
No, the limit is as given in the book, e1.
 
  • #7
ehild said:
No, the limit is as given in the book, e1.
My mistake -- I can't read my own writing...
 

1. What are exponential derivatives?

Exponential derivatives are a mathematical concept used in calculus to measure the rate of change of exponential functions. They are used to find the slope of a curve at a specific point on the graph.

2. How do you find the derivative of an exponential function?

To find the derivative of an exponential function, you can use the power rule or the logarithmic differentiation method. The power rule states that the derivative of ax is equal to axln(a). The logarithmic differentiation method involves taking the natural logarithm of both sides of the function and using the properties of logarithms to simplify the expression.

3. What is the significance of exponential derivatives?

Exponential derivatives are important in many fields, including finance, physics, and biology. They are used to model growth and decay processes, such as population growth, radioactive decay, and compound interest. They also have applications in optimization problems and differential equations.

4. Are there any rules for solving exponential derivatives?

Yes, there are several rules for solving exponential derivatives, including the power rule, the product rule, and the chain rule. These rules help simplify the process of finding the derivative of more complex exponential functions.

5. How can I apply exponential derivatives in real life?

Exponential derivatives can be applied in various real-life situations, such as calculating compound interest on investments, predicting population growth, and modeling the decay of radioactive substances. They can also be used in engineering and physics to analyze exponential systems and make predictions about their behavior.

Similar threads

Replies
1
Views
627
  • Calculus and Beyond Homework Help
Replies
1
Views
582
  • Calculus and Beyond Homework Help
Replies
29
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • STEM Educators and Teaching
Replies
22
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
995
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
28
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
Back
Top