Property derivative

Introduction: Recall that the derivative of a function $y(x)$, denoted by $y'(x)$, is defined as
$$\dfrac{y(x+\Delta x) - y(x)}{\Delta x} \rightarrow y'(x) \quad\text{if}\quad \Delta x \rightarrow 0.$$

Property: If $\Delta x$ is small, then
$$\dfrac{y(x+\Delta x) - y(x)}{\Delta x} \approx y'(x),$$
where the $\approx$-sign indicates that the left and right-hand side are approximately equal to each other. If we multiply the left and right-hand side by $\Delta x$, then we obtain
$$y(x+\Delta x) - y(x)\approx y'(x)\Delta x \qquad\text{or}\qquad \Delta y \approx y'(x)\Delta x.$$
The left-hand side is now the change of the function value and hence, this is approximately equal to the derivative multiplied by the change in the input variable.