StellarThe derivative of a function at a point is the instantaneous rate of change of the function at that point. It is also the slope of the tangent line to the graph at that point. Formally, the derivative is defined as a limit of average rates of change. If f is a function, then the derivative at x=a is when that limit exists. This formula is called the difference quotient. It measures the slope of a secant line over a very small interval of width h, and then the limit turns that secant slope into the exact tangent slope.
There are several ways to write derivative notation, and students must become comfortable reading all of them. Common notations include f'(x), y', dy/dx, and d/dx*(f(x)). These all represent the derivative, though the context determines whether the derivative is being viewed as a number at a point or as a new function. The derivative is one of the most important ideas in all of calculus because it turns the intuitive notion of “how fast something is changing right now” into a precise mathematical object.