Source: algebrica.org - CC BY-NC 4.0 https://algebrica.org/differential-of-a-function/ Fetched from algebrica.org post 6161; source modified 2025-05-17T16:55:19.
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 $\Delta x$:
The differential of the independent variable $x$ is equal to the increment of the variable itself: $\mathrm{d}x = \Delta x.$ By substituting the value into the definition, we obtain:
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:
From a geometric point of view, consider the triangle ABC. By the properties of trigonometry and of right triangles, the side $\overline{BC}$ can be rewritten as:
where $\overline{AB} = \Delta x$ and $\tan(\alpha) = f’(x)$. The equality $(4)$ can therefore be rewritten as:
In other words, the differential $dy$ 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 $x + \Delta x$.