Science:Math Exam Resources/Courses/MATH102/December 2011/Question 01 (a) ii
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Question 01 (a) ii 

ShortAnswer Questions. A correct answer in the box gives full marks. For partial marks work needs to be shown. Find the derivative of
f(x)=\frac{x}{\ln(\frac{1}{x})} 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Try to run through an algorithm every time you compute a derivative. Do you just know the derivative? If not what rules can you apply in this situation? Notice that you're dividing two functions and that the bottom function is a composition of two functions. 
Hint 2 

The quotient rule reads if then . The chain rule reads if then . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. This is a quotient of functions and so let and . Then, by the quotient rule, we have
Now, and for , we use the chain rule to see that
Note, we could also have noticed that and hence . Thus, combining all this information, we have
and any of the last three answers would be accepted. 