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Over the summer for AP Calculus homework we were given a packet of problems to figure out how to do in preparation for next year. One of the questions was to find the derivative of the function f(x)=(x^(1/2))+(x^(1/3))+(x^(2/3)). I simplified the derivative down to (1/(2sqrt(x))) + (1/(3(x^(2/3)))) + (2/(3(x^(1/3)))), but I am stuck. I don't know how to simplify it into one single fraction. Any help would be greatly appreciated. Thanks!

The expression you give for the derivative is correct and is a direct application of a simple rule for derivatives (and as such is not really simplified). I'm not sure why you need to transform the answer into a single fraction. The expression can be manipulated in various ways but I'm not sure myself how to turn it into a single fraction. Here's one transformation

f'(x) = (1/sqrt(x))( (1/3)x^(-1/6) + 1/2 + (2/3)x^(1/6) )

which has sort of a nice symmetry, but I don't know that it reveals much.

Please let me know if there is more to this problem.

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