My English lesson The derivative function BE CAREFUL!! B DEFINITION F ease of writing, we will often For express the function y = f(x) by f and its derivative function by D(f) or f . Given the function y = f(x), let us define the derivative of the function f (x), we denote by D(f(x)), or by y = f (x), which provides the values of the angular coefficient of the tangents to the graph of the function f(x). Let us look at some characteristics with the following example. example O A function y = f(x) is assigned (on the left) through its graph. To its right, the graphs of two other functions are drawn, one of which is the derivative f of the given function. By analysing the stationary points of f(x) and the intervals on which it is increasing or decreasing, determine which of the two graphs to the right is the graph of its derivative function. 2 y y y 1 1 2 O 2 x 2 y = f(x) O 2O x 2 2 x II. I. Let us simultaneously analyse what the growth/decrease of f(x) is and what the sign of the derivative should be, and report the results in a table: x 2 f decreasing stationary increasing stationary decreasing stationary increasing D(f) 0 =0 0 The derivative of f(x) is then represented in graph I. The primitive function We know that if y = f(x) is continuous and defined in an interval and its derivative function y = f (x) exists in that interval, this provides the values of the angular coefficients of the tangents to the graph of the function. But, for all k R, the graph of the function y = f(x) + k has the same altimetric trend as that of y = f(x), being simply translated with respect to it along vector v = (0 ; k). All functions drawn on the left, in the graph below, have as their derivative function the one drawn in the graph on the right. At each value of x, therefore, the angular coeffif cient of the tangent to the graph of y = f(x) is equal to the angular coefficient of the tangent to the graph of y = f(x) + k. We have, for all k R: D(f(x)) = D(f(x) + k). f x4 x1 228 x2 x3 x1 x2 x3 x4 If, therefore, y = m(x) is the derivative of a function, there are then an infinite number of functions of which it is the derivative function.

My English lesson