f(x) is a piecewise function defined as: f(x) = (3/2)x when 0 <= x <= 2 or f(x) = -(3/2)x+6 when 2 < x <= 4
g(x) is defined as: g(x) = -(1/4)x+1 when 0 <= x <= 4
Based on the graph or through the equations we can say: f(1) = (3/2)*1 = 1.5 g(1) = -(1/4)*1+1 = 0.75 f(3) = (3/2)*3 = 4.5 g(3) = -(1/4)*3+1 = 0.25
And the derivative values are: f ' (1) = 3/2 = 1.5 f ' (3) = -3/2 = -1.5 g ' (1) = -1/4 = -0.25 g ' (3) = -1/4 = -0.25 which are the slopes of each line
So... h(x) = f(x)*g(x) h ' (x) = f ' (x)*g(x) + f(x)*g ' (x) ... product rule h ' (1) = f ' (1)*g(1) + f(1)*g ' (1) h ' (1) = 1.5*0.75 + 1.5*(-0.25) h ' (1) = 0.75
and, h(x) = f(x)*g(x) h ' (x) = f ' (x)*g(x) + f(x)*g ' (x) h ' (3) = f ' (3)*g(3) + f(3)*g ' (3) h ' (3) = -1.5*0.25 + 4.5*(-0.25) h ' (3) = -1.5
