What is the mistake in my derivative calculation and how can I fix it?

[quasar987]

(And you will be greatly rewarded in the afterlife)

It's crazy, I don't see where I goes wrong. Lookit..
$$\frac{\partial}{\partial x} \left(\frac{x}{z\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2}\right)} \right)$$
$$= \left(\frac{1}{z\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2}\right)} \right)$$
$$+(x/z)(-1)\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2}\right)^{-2}\left((1/2)(x^2+y^2)^{-1/2}2x+(1/z^2)(3/2)\sqrt{x^2+y^2}2x\right)$$
$$= \left(\frac{1}{z\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2}\right)} \right)$$
$$-\frac{3x^2\sqrt{x^2+y^2}}{z^3\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2}\right)^2}$$
$$-\frac{x^2}{z\sqrt{x^2+y^2}\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2} \right)^2}$$

The first term agrees with Mapple but not the second and third.
In the second, the 3 is a 2 and in the third, the big parenthesis is not raised to the 2. (it's to the 1)

I have checked using the command 'simplify' that our two expressions are not equivalent. what Mapple does is that it get a (x²+y²)^½ out of the parenthesis before taking the derivative. And the worst thing is, when I do that too, I get the same result as mapple, but not when I don't take out (x²+y²)^½ first. I've banged my head on the desk for hours on this and get's see why I don't get the same answer by the two "methods". Help me obi-wan kenobi. You are my only hope.

 

[VietDao29]

$$-\frac{x^2}{z\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2} \right)^2}$$

It seems that this term is wrong. I get:
$$- \frac{x ^ 2}{z\sqrt{x ^ 2 + y ^ 2} \left( \sqrt{x ^ 2 + y ^ 2} + \frac{\sqrt{x ^ 2 + y ^ 2} ^ 3}{z ^ 2} \right) ^ 2}$$
*You seem to forget a $\sqrt{x ^ 2 + y ^ 2}$ in the denominator...*
You should check it again

 

[benorin]

Here's what I got

For that term, I get $$-\frac{x^2}{z\sqrt{x^2+y^2}\left(\sqrt{x^2+y^2} +\frac{\sqrt{x^2+y^2}^3}{z^2} \right)^2}$$

When I simplify that term I get $$-\frac{x^2}{\left( x^2+y^2\right)^{\frac{3}{2}} \left( x^2+y^2+z^2\right)^2}$$


I attached a Maple worksheet with my work (I use Maple v9.51) rename the file without the .txt so it ends with .mw (PF wouldn't let me upload .mw files, but .txt OK).

 

[quasar987]

VietDao: you'Re right, but it was just a Latex error. I have that too and haver edited the original post.

I still don't see where my mistake is.
Here's what my mapple has to say: