The concept is not a difficult one. If you have a function, say $f(x,y,z)=x{y}^2{z}^3$ from a point in space $(x,y,z)$ to a real number $f(x,y,z)$ then the partial derivatives with respect to $x$, $y$ and $z$ are just what you get when you treat the other variables as constants, so

$\partial f/\partial x=y^2{z}^3$

$\partial f/\partial y=2xy{z}^3$

$\partial f/\partial z=3x{y}^2{z}^2$

And that's it!

It has the important property that you can change the order (check this for the example above)

$\partial /\partial x(\partial /\partial y)f=\partial /\partial y(\partial /\partial x)f$

And if $x$, $y$, $z$ are functions of $t$, so

$f(x(t),y(t),z(t))$

then the total derivative of $f$ is

$\mathrm{df}/\mathrm{dt}=\partial f/\partial x\mathrm{dx}/\mathrm{dt}+\partial f/\partial y\mathrm{dy}/\mathrm{dt}+\partial f/\partial z\mathrm{dz}/\mathrm{dt}$

As an exercise, let $x(t)=t$, $y(t)=t^2$, $z(t)=t+1$ and use this in the pervious example for $f$ to calculate the total derivative $\mathrm{df}/\mathrm{dt}$.