Science:Math Exam Resources/Courses/MATH110/December 2011/Question 11
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Question 11 

Find a piecewise algebraic expression for the n^{th} derivative of
Note: This question was for bonus marks, so it may be more challenging. 
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? 
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Hint 

Write out the first few derivatives of . What pattern do you notice? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We compute the first few derivatives of and look for a pattern. After the second derivative, we notice that the derivative is a fraction, with a constant in the numerator and a power of x in the denominator. In fact, we can write this pattern as for all n ≥ 2. The only derivative that is different is the first one. Thus, our piecewise function describing the derivative is:
\ln(x) & n = 1 \\ \frac{ (n2)! (1)^n }{x^{n1}} & n > 1 \end{cases} 