Derivative and Differential - A Geometric Point of View
Derivative
Given a function and a point on it, if we increment the position of by a quantity , we obtain the corresponding point on the function graph.
For all increments of the independent variable, the corresponding increment of the dependent variable is .
is the difference quotient of the function, and it is equal to the trigonometric tangent of the angle that the line creates with the positive direction of the x-axis (i.e. the slope of the line).
The derivative of the function at point , is defined as the limit for of the difference quotient.
But when the increment tends to 0, the line tends to the tangent line to the graph of the function at , therefore the derivative of a function at a point is the slope of the tangent line to the graph of at point .
Differential
Given a function , differentiable at a point , the differential of the function is defined as , that is the product of the derivative of the function at that point by the infinitesimal increment of the independent variable.
For this definition we are using the notation instead of to denote infinitesimal increments.
If you want to learn more about the history of this notation, you can start from here.
The differential is the measure of the increment of the y-coordinate of a point on the tangent line to the graph of the function, corresponding to the infinitesimal increment of its x-coordinate.
Explore Similarities and Differences
Food for Thought
The general equation of the tangent line to the graph of a function at a point is . Replace with and simplify. What is the right hand side of the equation? What is the left hand side of the equation? Describe the connection between the algebraic expression on the right hand side and the geometric representation of the expression on the left hand side of the equation.