How do you find the derivative of #3x^2 (4x - 12)^2 + x^3 (2) (4x - 12) (4)#?

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Answer 1 · 2015-08-23T21:47:43

Rewrite a bit and then you have a choice.

#3x^2 (4x - 12)^2 + x^3 (2) (4x - 12) (4) = 3x^2 (4x - 12)^2 + 8x^3(4x - 12)#

Now we could use the product rule to differentiate each of the two terms.

I prefer to factor first.

#3x^2 (4x - 12)^2 + 8x^3(4x - 12) = x^2(4x-12)[3(4x-12)+8x]#

# = x^2(4x-12)(20x-36)#

# = 16x^2(x-3)(5x-9)#

Now we can either distrubute the #16x^2# to one factor and use the usual product rule, or we can use the three factor product rule:

#d/dx(uvw) = u'vw+uv'w+uvw'#

So we get:

#32x(x-3)(5x-9)+16x^2(1)(5x-9)+16x^2(x-3)(5)#

This can also be factored and simplified if you wish.