Stuck on partial derivatives (again :( )

Okay,
I have uploaded a picture of the equation that I'm struggling with...


f'x (x,y) is (1 + x^2 + y^2 ) ^ -1/2 x is apparently the answer

I'm really confused by this as I don't understand why it became negative or why an x is added at the end, I had just treated it as a constant.

Can anyone out there help ?

Thanks for reading the long post xx

Okay,
I have uploaded a picture of the equation that I'm struggling with...


f'x (x,y) is (1 + x^2 + y^2 ) ^ -1/2 x is apparently the answer

I'm really confused by this as I don't understand why it became negative or why an x is added at the end, I had just treated it as a constant.

Can anyone out there help ?

Thanks for reading the long post xx

When you differentiate with respect of x then it is x that is no constant
The answer you wrote is not completed derivative of f(x,y) because
contains the derivative of only the first part of f(x,y).
THe first part of f(x,y) is a composite function composed from:
1. $\displaystyle u(x)=\sqrt{x}$
2. $\displaystyle v(x)=1+x^2+y^2$
So
$f(x)=u\circ v=\sqrt{1+x^2+y^2}$
For differentiating use the chain rule
$\displaystyle f'_x(x,y)=u'_v \circ v \cdot v'_x=\frac{1}{2}\cdot \left (\sqrt{1+x^2+y^2}\right )^{-\frac{1}{2}} \cdot 2x$

You can see why there is an x as factor