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

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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...

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

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

$= 36x - 24 + 36x + 6$

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

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