What is the differences between differentiation and partial differentiation like...

\(\displaystyle z = x^2 + y^2\)

find ∂z/∂x and dz/dx??? Im confused with this two...

Partial differentiation means finding how one variable changes with respect to another one variable, with no other variables making any difference.

So in this case, if you were finding \(\displaystyle \frac{\partial z}{\partial x}\), that would mean you are asked to describe how \(\displaystyle z\) changes from changes in \(\displaystyle x\). Note that \(\displaystyle y\) will not have any effect, and so gets treated as constant.

If you treat \(\displaystyle y\) as being constant, then

\(\displaystyle \frac{\partial z}{\partial x} = 2x\) (because the derivative of the constant \(\displaystyle y\) is 0).

The only way you could find \(\displaystyle \frac{dz}{dx}\) is if you knew that \(\displaystyle y\) was some function of \(\displaystyle x\). If it was, then

\(\displaystyle \frac{dz}{dx} = \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2)\)

\(\displaystyle = 2x + \frac{d}{dy}(y^2)\,\frac{dy}{dx}\)

\(\displaystyle = 2x + 2y\,\frac{dy}{dx}\).