Differential of a Function
Consider $f ( x )$ a differentiable function on the interval $[ a , b ]$. Since the function is differentiable, it is also continuous on the given interval. Let us consider two points $x$ and $x + \Delta x \in [ a , b ]$.
It is defined the differential of a function $f ( x )$, relative to the point $x$ and the increment $\Delta x$, as the product of the derivative of the function evaluated at $x$ and the increment $\Delta x$:
\[(\text{1}) d y = f^{'} ( x ) \cdot \Delta x\]The differential of the independent variable $x$ is equal to the increment of the variable itself: $d x = \Delta x .$ By substituting the value into the definition, we obtain:
\[(\text{2}) d y = f^{'} ( x ) \cdot d x\]From the formula, it follows that the first derivative of a function is the ratio between the differential of the function and that of the independent variable:
\[(\text{3}) f^{'} ( x ) = \frac{d y}{d x}\]
From a geometric point of view, consider the triangle ABC. By the properties of trigonometry and of right triangles, the side $\overset{―}{B C}$ can be rewritten as:
\[(\text{4}) \overset{―}{B C} = \overset{―}{A B} \cdot tan ( \alpha )\]where $\overset{―}{A B} = \Delta x$ and $tan ( \alpha ) = f^{‘} ( x )$. The equality $( 4 )$ can therefore be rewritten as:
\[(\text{5}) \overset{―}{B C} = \overset{―}{A B} \cdot tan ( \alpha ) \\ & = \Delta x \cdot f^{'} ( x ) \\ & = d y\]In other words, the differential $d y$ is the change in the ordinate of the tangent line to the curve when moving from point A with abscissa $x$ to point B with abscissa $x + \Delta x$.