AS/A2 Pure 1 Pure 2

P3 Topic 4: Differentiation
Differentiating a product
The function x2(x-1) is a product of x2 and (x-1). The derivative of x2 is 2x and the derivative of x-1 is 1. So, is the derivative of x2(x-1) equal to 2x(1) = 2x, or 2x + 1, or something else? The display shows the curve y = x2(x-1) and its gradient function y = f'(x). Clearly the function f'(x) is neither 2x nor 2x+1. What is it? Work out also the gradient functions for some of the other examples, particularly (B) x(x+1) and (C) x2(x2-1) .

Summary The gradient function of x2(x-1) is 3x2-2x The gradient function of x(x+1) is 2x+1 The gradient function of x2(x2-1) is 4x3-2x The gradient function of xsin x is xcos x + sin x The gradient function of xcos x is -xsin x + cos x The gradient function of xex is xex+ex The gradient function of xln x is 1+ln x The gradient function of exsin x is excos x+exsin x Based on free Java applets from JavaMath