The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment h

The derivative of a function f ( x ) at a value a is found using either of the definitions for the slope of the tangent line As such, the velocity v ( t ) at time t is the derivative of the position s ( t ) at time t .

derivative — the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative. instantaneous rate of change — the rate of change of a function at any point along the function a , also called f ′ ( a ) , or the derivative of the function at a. constant multiple rule — the derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative: d x ( c f ( x ) = c f ′ ( x ). derivative function — gives the derivative of a function at each point in the domain of the original function for which the derivative is defined.