Simple derivation I just can't get

  • Thread starter hks118
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    Derivation
In summary, the student attempted to find the derivative of g(x) using different methods but all of them yielded incorrect results. He eventually found the derivative using the chain rule and proved that it was 1/x + (1/2)*1/(x2 - 11)*d/dx(x2 - 11).
  • #1
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Homework Statement


g(x)=ln(x*sqrt(x2-11)


Homework Equations


d/dx lnx=1/x


The Attempt at a Solution


I've attempted a few solutions:
1/(x*sqrt(x2-11)
(1/x)+1/(sqrt(x2-11)
etc

but they have been wrong. I'm not sure how to differentiate a ln of a product...thats why I tried to break it into lnx+ln sqrt(x2-11
 
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  • #2
You were correct in how you began to differentiate g(x), but after you use D[ln(x)]=1/x, you must apply the chain rule. That is, you must multiply by x*sqrt(x2-11)
 
  • #3
Actually, while that rule is correct, it's incomplete.

[tex]\frac{d}{dx}ln\left(f(x)\right)=\frac{f'(x)}{f(x)}[/tex]

This works even for ln(x) since the derivative of x is 1, so nothing changes.
All you need to do is multiply everything by the derivative of what's inside the log function.
 
  • #4
hks118 said:

Homework Statement


g(x)=ln(x*sqrt(x2-11)


Homework Equations


d/dx lnx=1/x


The Attempt at a Solution


I've attempted a few solutions:
1/(x*sqrt(x2-11)
(1/x)+1/(sqrt(x2-11)
etc

but they have been wrong. I'm not sure how to differentiate a ln of a product...thats why I tried to break it into lnx+ln sqrt(x2-11
What you started doing here would have worked if you had completed it. Also, for future reference, your problem statement should say what it is that your are trying to do. You didn't make it explicit that you wanted to find the derivative of g(x). The word "derivation" does not imply that you are taking the derivative -- the word for that is differentiation.

One other thing: you started with an equation -- g(x) = ln(x*sqrt(x2-11) -- each step should have been an equation.

So, you started with
g(x) = ln(x*sqrt(x2-11) = ln(x) + ln(sqrt(x2 - 11) = ln(x) + (1/2)ln(x2 - 11)
==> g'(x) = d/dx(ln(x)) + d/dx[(1/2)ln(x2 - 11)) = 1/x + (1/2)*1/(x2 - 11)*d/dx(x2 - 11)
==> g'(x) = 1/x + 2x/(2(x2 - 11)) = 1/x + x/(x2 - 11)

In the 2nd line I used the fact that d/dx(kf(x)) = k*d/dx(f(x)) = k*f'(x).
I also used the chain rule form of the derivative of the natural log function - d/dx(ln(u) = 1/u * du/dx.
 
  • #5
Thanks guys! Finally got the right answer with your help. I'll remember to format my question the right way next time too,
 

1. What is a simple derivation and why is it important?

A simple derivation is a logical and step-by-step process of obtaining a result or equation from known information or principles. It is important because it allows scientists to understand and explain complex phenomena in a clear and concise manner.

2. How do I approach a simple derivation?

The first step in approaching a simple derivation is to clearly define the problem and identify what information and principles are known. Then, you can use logic and basic mathematical operations to work towards the desired result.

3. What if I get stuck during a simple derivation?

If you get stuck during a simple derivation, it is important to take a step back and review the information and principles you are using. You can also try breaking the problem into smaller, more manageable steps or seeking help from a colleague or mentor.

4. Can a simple derivation be wrong?

Yes, a simple derivation can be wrong if there are errors in the logic or if incorrect information is used. It is important to double check your work and make sure all steps are clearly explained and supported by the known information.

5. How can I improve my skills in simple derivation?

Practice and repetition are key to improving your skills in simple derivation. It is also helpful to study and understand the underlying principles and equations involved in the derivation process. Seeking feedback and guidance from experienced scientists can also aid in improvement.

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