Given a function [math]y=f\left(x\right)[/math] and a point [math]P=\left(x,f\left(x\right)\right)[/math] on it, if we increment the position of [math]x[/math] by a quantity [math]\Delta x[/math], we obtain the corresponding point [math]Q=\left(x+\Delta x,f\left(x+\Delta x\right)\right)[/math] on the function graph.[br][br]For all increments [math]\Delta x[/math] of the independent variable, the corresponding increment of the dependent variable is [math]\Delta y=f\left(x+\Delta x\right)-f\left(x\right)[/math].[br][br][math]\frac{\Delta y}{\Delta x}=\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}[/math] is the difference quotient of the function, and it is equal to the trigonometric tangent of the angle that the line [math]PQ[/math] creates with the positive direction of the [i]x[/i]-axis (i.e. the slope of the line).[br][br]The derivative of the function [math]y=f\left(x\right)[/math] at point [math]x[/math], is defined as the limit for [math]\Delta x\rightarrow0[/math] of the difference quotient.[br][br]But when the increment [math]\Delta x[/math] tends to 0, the line [math]PQ[/math] tends to the tangent line to the graph of the function at [math]x[/math], therefore [i][color=#1e84cc]the derivative of a function at a point is the slope of the tangent line[/color][/i] to the graph of [math]y=f\left(x\right)[/math] at point [math]x[/math].

Given a function [math]y=f\left(x\right)[/math], differentiable at a point [math]x[/math], the differential of the function is defined as [math]dy=f'\left(x\right)\cdot dx[/math], that is the product of the derivative of the function at that point by the infinitesimal increment of the independent variable.[br][br]For this definition we are using the notation [math]dx[/math] instead of [math]\Delta x[/math] to denote infinitesimal increments. [br]If you want to learn more about the history of this notation, you can start from [url=https://en.wikipedia.org/wiki/Leibniz%27s_notation]here[/url].[br][br]The [i][color=#1e84cc]differential [/color][/i]is the measure [math]dy[/math] of the [i][color=#1e84cc]increment of the y-coordinate of a point on the tangent line[/color][/i] to the graph of the function, corresponding to the infinitesimal increment [math]dx[/math] of its [i]x[/i]-coordinate.[br][br]