## Derivative of cubed root of x with limits

Limits, differentiation, related rates, integration, trig integrals, etc.
Andromeda
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Joined: Tue Oct 12, 2010 1:56 am
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### Derivative of cubed root of x with limits

I am having trouble finding the derivative of cubed root of x using only limits. I know it is easy to use chain rule but I cannot use that yet.

stapel_eliz
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I am having trouble finding the derivative of cubed root of x using only limits.
What expression, exactly, are you working with (the cube of the square root? the cube of the fifth root? something else?)? What have you tried so far?

Andromeda
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Joined: Tue Oct 12, 2010 1:56 am
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### Re: Derivative of cubed root of x with limits

I'm sorry, I meant the cube root of x. as in cuberoot(x).

So far a tried finding the derivative with a limit as h approaches 0
lim h-->0[(cuberoot(a+h) - cuberoot(a)) / ((a+h) - a)]

With this, I always end up with an indeterminate form.

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
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### Re: Derivative of cubed root of x with limits

I'm sorry, I meant the cube root of x. as in cuberoot(x).

So far a tried finding the derivative with a limit as h approaches 0
lim h-->0[(cuberoot(a+h) - cuberoot(a)) / ((a+h) - a)]

With this, I always end up with an indeterminate form.

$\lim_{h\to0}\frac{\sqrt[3]{a+h}-\sqrt[3]{a}}{h}$

use the fact that

$x^3-y^3=(x-y)(x^2+xy+y^2)$

Andromeda
Posts: 11
Joined: Tue Oct 12, 2010 1:56 am
Contact:

### Re: Derivative of cubed root of x with limits

I'm sorry, I meant the cube root of x. as in cuberoot(x).

So far a tried finding the derivative with a limit as h approaches 0
lim h-->0[(cuberoot(a+h) - cuberoot(a)) / ((a+h) - a)]

With this, I always end up with an indeterminate form.

$\lim_{h\to0}\frac{\sqrt[3]{a+h}-\sqrt[3]{a}}{h}$

use the fact that

$x^3-y^3=(x-y)(x^2+xy+y^2)$
How would that work since $sqrt(a+h)$ and $sqrt(a)$ are not cubed numbers?

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Derivative of cubed root of x with limits

I'm sorry, I meant the cube root of x. as in cuberoot(x).

So far a tried finding the derivative with a limit as h approaches 0
lim h-->0[(cuberoot(a+h) - cuberoot(a)) / ((a+h) - a)]

With this, I always end up with an indeterminate form.

$\lim_{h\to0}\frac{\sqrt[3]{a+h}-\sqrt[3]{a}}{h}$

use the fact that

$x^3-y^3=(x-y)(x^2+xy+y^2)$
How would that work since $sqrt(a+h)$ and $sqrt(a)$ are not cubed numbers?
let $x=\sqrt[3]{a+h}$ and $y=\sqrt[3]{a}$