Limit Definition of the Derivative

The slope of the line tangent to a curve can be represented by the equation of the derivative.

The derivative is one of the fundamentals of calculus; it shows how a function changes as its input changes. It can be thought of as the slope of the tangent line on a graph on the xy plane, or as the velocity of an object if the position of the object has a relationship to the time, t. The derivative (f'(x)) can be defined as the slope of a secant as the difference in the x values of the endpoints of the secant approaches 0 (i.e. f'(x) = lim(h->0) (f(x + h) - f(x))/h).  For example, the derivative of f(x)=x is lim(h->0) (f(x + h) - f(x))/h) = lim(h->0) ((x + h - x)/h) = lim(h->0) (h/h) = 1.