# Differential of a Function
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](../derivatives.md) function on the interval \\(\[a,b\]\\). Since the function is differentiable, it is also [continuous](../continuous-functions.md) 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\\):
\\[
\\mathrm{d}y = f’(x) \\cdot \\Delta x \\tag{1}
\\]
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:
\\[
\\mathrm{d}y = f’(x) \\cdot \\mathrm{d}x \\tag{2}
\\]
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:
\\[
f’(x) = \\frac{\\mathrm{d}y}{\\mathrm{d}x} \\tag{3}
\\]
From a geometric point of view, consider the triangle ABC. By the properties of [trigonometry](https://algebrica.org/trigonometry/) and of right triangles, the side \\(\\overline{BC}\\) can be rewritten as:
\\[
\\overline{BC} = \\overline{AB} \\cdot \\tan(\\alpha) \\tag{4}
\\]
where \\(\\overline{AB} = \\Delta x\\) and \\(\\tan(\\alpha) = f’(x)\\). The equality \\((4)\\) can therefore be rewritten as:
\\[
\\begin{align} \\overline{BC} &= \\overline{AB} \\cdot \\tan(\\alpha) \\tag{5} \\[0.5em] &= \\Delta x \\cdot f’(x) \\[0.5em] &= \\mathrm{d}y \\end{align}
\\]
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 \\).
