(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

original equation: f(x) = x[itex]^{2/3}[/itex]e[itex]^{x}[/itex]

first derivative: f'(x) = e^x[(3x+2)/3x^(-1/3)]

second derivative f"(x) = e^x[(9x^2 + 12x - 2)/9x^(4/3)]

2. Relevant equations

So far, I'm right here in the second derivative:

f"(x) e^x[(9x^2/3+12x^(-1/3)-2x^(-4/3)]

But I don't know how to get it into this final form:

f"(x) = e^x[(9x^2 + 12x - 2)/9x^(4/3)]

9x4=3

I am thinking I need to factor out an x to some power, but I don't know which power.

I know I'm supposed to factor out the smallest exponent, but does that mean the absolute smallest exponent, or the smallest in terms of the number furthest to the left on the number line?

In otherwords, do I factor out x^(-1/3) or x^(-4/3) ? And by factoring out one of these terms, am I heading in the right direction to that final second derivative form?

EDIT:

I should specify that my question is in regard to getting from the first derivative to the second derivative. That's where I'm getting stuck. I can get the derivative, but not into the simplified I posted.

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# Derivative of e^x[(3x+2)/3x^(1/3)]

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