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# Calculus/Math question

Question
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!

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

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Calculus

Volunteer

#### 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