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Question 02 (b) 

Suppose z = f(x,y) has continuous second order partial derivatives and x = r cos(t), y = r sin(t). Express the following partial derivatives in terms of r,t and the partial derivatives of f. b) 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

You should use your answer from part (a) to solve this problem: 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. To evaluate this derivative we merely need to recognize that we apply the first derivative operator to our answer from part (a): We need to apply the chain rule again as in part (a): where we have use the fact that x = r cos(t), y = r sin(t) to write the last equation above. From part (a), we found that Using this, we continue to evaluate the partial derivatives above finally giving us Writing the result above in terms of r, t gives the final answer: 