Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 04 (a)
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Question 04 (a) 

(a) Let . Calculate at . 
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 

Use implicit differentiation. 
Hint 2 

Differentiate both sides with respect to , and treat as a function of ; . 
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. As in Hint 2, we differentiate both sides with respect to . That means we treat as a function of , namely . For the left hand side, we use the chain rule with , , , and to yield For the right hand side we use the quotient rule with , , and to yield Therefore, we have To find at , we put and in the above expression: Since , the above equation simplifies to We solve this equation in the unknown : Answer: . 