Week 9 - Calculus Problem

Greeting Again!  This week I have a special treat!  I will write up my problem solving episode with the last question from my calculus exam.  To be honest I didn't get perfect on it because I completely forgot how to eliminate factorials, but I did get 7/8.  Not a bad result considering most people probably got close to 0 because it was marked strictly.

Question:

Understanding the Problem:
Goal: Take the 35th derivative of this function.
Input: Function
Process: Differentiation, but there must be a pattern to recognize due to time constraints.
Output: Function, by inspection I suspect the 35th derivative will involve factorials because the exponent on some of the x variables is 35 if you were to expand the numerator.

Devise A Plan:
First I will begin by simplifying the expression as much as possible, and putting it in a format that will make it easy to differentiate.  I will start by expanding the denominator and seeing if it can be simplified.  Ideally I will eliminate the division because differentiating + or - is much easier than x or /, but worst case I will rewrite the fraction as a product and rewrite the denominator as a term to a negative exponent.  From this point I will try to collect like terms if possible.

Once I have the original function simplified as much as possible I will begin to differentiate.  It is guaranteed that there is a pattern given time constraints.  Also I remember similar examples from class that used this pattern technique.  I will keep taking derivatives until I recognize said pattern, and then apply the pattern to figure out the 35th derivative.  At this point I will have solved the question.

Executing the Plan:

Once I got to the final line in step 1 I knew the function was simple enough to begin differentiating.  The first derivative helped me solve the last two terms of the 35th derivative because I knew the x raised to the power 35 would just end up equaling 35! because the power decreased once per differentiation.  Furthermore I knew the x raised to the power 34 would be 0 because the 34th derivative of this would be constant, so when you differentiated again it would be 0.  I took the second derivative to help me figure out how to solve the first term.  Because the exponent was negative it would get more and more negative each differentiation.  Also the term in from of (x+3) would alternate between negative and positive, being negative on odd differentiations and positive on even differentiations.  Since 35 is odd I knew that it must be negative.  Finally I knew the number at the front would have some sort of factorial, but not go all the way down to 1 because it started at 3.  So in order to eliminate the x2x1 you divide by 2!.

Looking Back
The key to questions like this really is to not lose your cool.  Any type of problem like this has to have some pattern in differentiation and their has to be a trick to simplifying the function.  I could check the result by taking all 35 derivatives manually, but this is too time consuming especially for a test.  I do not think this question could have been done any differently given time constraints; using the product or quotient rule would have made it way to complicated.