Really simple derivative that I somehow can't comprehend

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Koalanet21

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I have tried everything, it's the middle of the night, I feel like this forum is my last resort.

The result of the derivative 1/(x^(2/3)) is -2/(3*x^(5/3))
This is something that I can easily check on an online calculator like Allmath.
answer.PNG
But for some reason, I don't understand the "5/3"
Even though I apply the power rule
(n)' = nx^(n-1)

But when I apply it, it's wrong

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

Please tell what's wrong, I have almost lost hope
 
Koalanet21 said:
I have tried everything, it's the middle of the night, I feel like this forum is my last resort.

The result of the derivative 1/(x^(2/3)) is -2/(3*x^(5/3))
This is something that I can easily check on an online calculator like Allmath.
View attachment 27859
But for some reason, I don't understand the "5/3"
Even though I apply the power rule
(n)' = nx^(n-1)

But when I apply it, it's wrong

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

Please tell what's wrong, I have almost lost hope
Click to expand...
You got the power rule correctly. In your problem

n = - 2/3........ then

n -1 = -2/3 -1 = (-2 -3)/3 = - 5/3 ................. this is what you have - and there is your "5/3"

If you have further question/s, please write back telling us exactly where you are having trouble.
 
Oh now that part I understand

But why there is a "minus" before the 2, in my "answer" I don't have it. Where does it come from?
Can you develop the whole calculus so that I see everything clearly (like I did, but mine was wrong)
 
[MATH]y = \dfrac{1}{x^{2/3}}= x^{-2/3} \implies \dfrac{dy}{dx} = - \dfrac{2}{3} * x^{\{{(-2/3) - 1}\}} = - \dfrac{2}{3} * x^{-5/3}.[/MATH]
 
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