Question

what is the derivative of
x^x2

Answer 1

is the second x raised to the 2?

Answer 2

i think its x(2x)

Answer 3

hmm, perhaps x^2x..

Answer 4

This is a bit tricky and requires a little bit of thinking.

Firstly, to clarify.. you wand \(\frac{d}{dx}x^{x^2}\)

Answer 5

Err want.

Answer 6

Right?

Answer 7

yeah, i think that's what he means, because had it been multilication he/she would have just put the two in front..

Answer 8

Are you going to solve this polpak? :)

Answer 9

Crazy! How's this look?
<a href=" http://content.screencast.com/users/ffeldon/folders/Jing/media/df54790d-7291-4b5e-be30-d944f1a0ea34/2011-04-22_1313.png "><img class="embeddedObject" src=" http://content.screencast.com/users/ffeldon/folders/Jing/media/df54790d-7291-4b5e-be30-d944f1a0ea34/2011-04-22_1313.png " width="596" height="653" border="0" /></a>

Answer 10

Sure. The original poster seems to have gone away, but it's a fun exercise in anycase.
Let \(u = x^{x^2} \implies (ln\ u) = x^2(ln\ x)\)
\[\frac{d}{dx}(ln\ u) = \frac{d}{dx}x^2(ln\ x)\]
\[\implies \frac{d}{du}(ln\ u) * \frac{du}{dx} = x^2*\frac{d}{dx}(ln\ x) + (ln\ x)*\frac{d}{dx}(x^2)\]
\[\implies \frac{1}{u} * \frac{du}{dx} = \frac{x^2}{x} + 2x(ln\ x) \]
\[\implies \frac{du}{dx} = u(1 + 2x(ln\ x))\]
\[ = x^{x^2}(1 + 2x(ln\ x)) = x^{x^2} + 2x^{x^2+1}(ln\ x)\]
\[\implies \frac{d}{dx}x^{x^2} = x^{x^2} + 2x^{x^2+1}(ln\ x)\]

Answer 11

i dont uderstand how you got to ln

Answer 12

I took the ln(u) to get the x^2 out of the exponent.

Answer 13

If \(u = x^{x^2}\) then \(ln(u) = x^2ln(x)\)

Answer 14

did you use a formula on the rest of it am kinda lost

Answer 15

No formula. Just the chain rule and the product rule.

Answer 16

The right side of the equation on line 3 is the chain rule, the left is the product rule.

Answer 17

you made a small error x^2/x=x not 1

Answer 18

Ah, so I did.

Answer 19

I had the 1 there because I was gonna factor the x from both terms, then forgot to do it ;p

Answer 20

the equation still holds if polak takes the natural log of both sides then you use implicit differentiation to solve for y'

Answer 21

so when trying to deferentiate x^2*lnx you use the product rule

Answer 22

yes