How do I write this in functional notation?

Let's say I have the following:

$u=t^4+2t$

$\frac{du}{dt} = 4t^3+2$

$du = (4t^3+2)dt$

Now, if I wanted to write this in functional notation:

$f(t) = t^4+2t$

$f'(t) = 4t^3+2$

What is the next step in functional notation?

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$\frac{du}{dt}=f'(t)\implies du=f'(t)dt$ :) –  Jonas Meyer Jan 4 '12 at 5:20
It really depends on what you mean by your original third line. Maybe try $f(t+h) - f(t) = ( 4 t^3 + 2) h + O(h^2)$? –  Scaramouche Jan 4 '12 at 5:21