How do you differentiate #f(x)=(2x-1)(3x-2)(5x+1)# using the product rule?
1 Answer
Well, you don't have to do it with the product rule, but I guess I can do it both ways.
Product Rule:
#f(x) = g(x)h(x)#
#f'(x) = g(x)h'(x) + h(x)g'(x)#
#f(x) = k*x^n#
#f'(x) = kn*x^(n-1)#
Easier way:
#f(x) = (6x^2 - 4x - 3x + 2)(5x + 1)#
#= 30x^3 + 6x^2 - 20x^2 - 4x - 15x^2 - 3x + 10x + 2#
#= 30x^3 - 29x^2 + 3x + 2#
#color(blue)(f'(x) = 90x^2 - 58x + 3)#
Here you just multiply everything out first and then use the power rule. Personally I find the power rule easier to remember via muscle memory.
Harder way:
#f(x) = (6x^2 - 4x - 3x + 2)(5x + 1)#
#color(blue)(f'(x)) = (6x^2 - 7x + 2)(5) + (5x + 1)(12x - 7)#
#= 30x^2 - 35x + 10 + (5x + 1)(12x - 7)#
#= 30x^2 - 35x + 10 + 60x^2 - 35x + 12x - 7#
#= color(blue)(90x^2 - 58x + 3)#
Here you partially multiply things out, and then use the product rule, and then continue multiplying things out.
