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Question:   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!

Answer
Hello Courteney,

Your answer is fine.  Why get a single fraction?  It will be somewhat complicated looking!
I'd leave the way you have it!

But, if you want...
The least common denominator is 6x^(2/3), so...
Multiply 1/[2x^(1/2)], by [3x^(1/6)]/[3x^(1/6)] to get [3x^(1/6)]/[6x^(2/3)]
Multiply 1/[3x^(2/3)] by 2/2 to get 2/[6x^(2/3)]
Multiply 2/[3x^1/3)] by [2x^(1/3)]/[2x^(1/3)] to get [4x^(1/3)]/[6x^(2/3}]

Now we can add then to get  [3x^(1/6)+2+4x^(1/3)]/[6x^(2/3)]

OK?  I hope this helps!

TTYL, Abe

Calculus

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Abe Mantell

Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook

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