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!
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!