How do you calculate the derivative of negative fractal exponents?

Question

I took brief calc 20 years ago in a summer session and what little I learned has long faded, so I've been trying to reteach myself.  Unfortunately, the first book I used to teach myself didn't explain how a particular problem's answer was reached, nor did the second book I bought on the subject.

here's the problem, when the exponent is a negative fractal, like -(2/3), the yielded derivatives are very different if you treat the exponent as (-2 * 1/3), (2* -1/3) or (-1*2*1/3).

which way do I go?  Why?

? · Answer

A couple of things: it may sound petty but there is a difference between fractal and fraction, Fraction is the term you should be using. I just wanted to straighten that out.

Secondly, I'm not sure what you are trying to show by breaking the fraction into products. It seems a bit confusing.

Anyway, if you have a function f of the form 
f(x) = x^a, where a is any real number, then the derivative is 
df/dx = ax^(a-1)

So, if a is a negative fraction, -2/3 say, then
f(x) = x^(-2/3) gives
df/dx = -(2/3)x^(-5/3)

That is the only answer. There are no others, no matter how you express the fraction.

? · Answer

the derivative of x^n is n x^(n-1) for ALL real numbers n. The derivative of x^(-2/3) is (-2/3) x^(-5/3).

Using the chain rule, the derivative of (x^(-2))^(1/3) is

(1/3) (x^(-2))^(-2/3) * (-2) x^(-3) =
(1/3)(-2) x^(4/3 - 3) =
(-2/3) x^(-5/3)

just as before.

MechEng2030 · Answer

Most books do give step by step reasoning. You have to apply d/dx[x^n] = n*x^(n - 1) rule on taking the derivative of x^(-2/3).

How do you calculate the derivative of negative fractal exponents?

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