How do you use the Product Rule to find the derivative of #f(x)=(6x-4)(6x+1)#?

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Answer 1 · 2015-07-17T21:18:44

#f'(x)=72x-18#

In general, the product rule states that if #f(x)=g(x)h(x)# with #g(x) # and #h(x#) some functions of #x#, then #f'(x)=g'(x)h(x)+g(x)h'(x)#.

In this case #g(x)=6x-4# and #h(x)=6x+1#, so #g'(x)=6# and #h'(x)=6#. Therefore #f(x)=6(6x+1)+6(6x-4)=72x-18#.

We can check this by working out the product of #g# and #h# first, and then differentiating. #f(x)=36x^2-18x-4#, so #f'(x)=72x-18#.

Answer 2 · 2015-07-17T22:11:38

You can either multiply this out and then differentiate it, or actually use the Product Rule. I'll do both.

#f(x) = 36x^2 + 6x - 24x - 4 = 36x^2 - 18x - 4#

Thus, #color(green)((dy)/(dx) = 72x - 18)#

or...

#d/(dx)[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)#

#= (6x-4)*6 + (6x+1)*6#

#= 36x - 24 + 36x + 6#

#= color(blue)(72x - 18)#