The previous example can be generalized as follows: Suppose the function $y=f(t)$ is differentiable in the point $t$, that is, has a non-vertical tangent line in this point. For a small increase ${\vartriangle}t$ of $t$, one can estimate the increase ${\vartriangle}y=f(t+{\vartriangle}t)-f(t)$ well by the following formula:

$${\vartriangle}y\approx f'(t)\cdot {\vartriangle}t$$ The closer in the neighbourhood of $t$, that is, the smaller ${\vartriangle}t$, the better is the estimated value of the function. The underlying idea of this formula is that the graph of a smooth function is in practice almost a straight line when one sufficiently zoomes in. For negligible changes, denoted with $dy$ and $dt$, the following statement can be written down:

The differential $f$ therefore depends on the time $t$ and the infinitesimal change $dt$. If $dy$ is denoted as $df$, the relationship between differentials and derivatives is easily made: the quotient of the differentials $df$ and $dt$, also known as differential quotient is equal to the derivative $f'(t)$ in $t$. Hence the mixed use of $y'$ and $\displaystyle\frac{dy}{dt}$ for the derivative function in this instructional material.

It is thus clear that a differential equation, that is, an equation in which besides a yet unknown function also one or more derivatives of that function are present can also be written as an equation between differentials: for example, the differential equation $\displaystyle\frac{dy}{dt}=y$ can be written as a relationship between differentials as $dy=y\,dt$. More generally, the differential equation $\displaystyle\frac{dy}{dt}=\varphi(t,y)$ with $\varphi$ a function of two variables, can be rewritten in terms of differentials as $dy=\varphi(t,y)\,dt$.