Science:Math Exam Resources/Courses/MATH200/April 2012/Question 03
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Question 03 

Let , where has continuous second order partial derivatives, and Find when t=1. 
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 

To evaluate the derivative of a multivariable function g = g(x,y) with respect to t where x,y are functions of t, we need to use the chain rule for multivariable functions: 
Hint 2 

After applying the multivariable chain rule to ƒ once, you will need to apply it again to dƒ/dt since the partial derivatives of ƒ are also functions of x, y. You will also need the product rule. 
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. First we evaluate the first derivative of z with respect to t using the multivariable chain rule. We then take the next derivative of z by taking the derivative with respect to t of the above result, using the multivariable chain rule again as well as the product rule. Next, we recognize that if t = 1, then (x,y) = (2,1). Thus, we can use the information given in the question about the partial derivatives of f to evaluate at t = 1: Therefore, 