Is there an easy way to find the derivative of this function:?

Question

F(x)= x^1/6 – 2x^-1/6 +5

The answer was supposed to be f ‘(x)= x^1/3+2 / (6x^7/6)

Answer 1

This is very easy.

F'(x) = (1/6) x^-5/6 - 2 (-1/6) x^-7/6 + 0
F'(x) = (1/6) x^-5/6 +(1/3) x^-7/6

I think you may have an error in your book.

d(x^1/6) / dx = (1/6) x^(1/6 - 6/6) = (1/6) x^(1 - 6) / 6 = (1/6) x^-5/6.

I can see how they got their answer for the last term. They simply moved the variable into the denominator and eliminated the negative exponent. However, doing the same thing to the first term will not result in their answer.

1 / (6 x^5/6) does not equal x^1/3 by any stretch of the imagination.