I am trying to prove certain characteristics of quadratic functions. In doing so, I need to find the derivative of
________________________f(x)=(x-a)(x-b)(x-c).
However I am stuck as how to apply the product rule to a function like this. At first I simply let each piece be 'u' 'v' and 'w' respectively, and applied u'vw+uv'w+uvw'. Notice that the derivative of each piece is 1, so:
_________________f '(x)=1(x-b)(x-c)+(x-a)1(x-c)+(x-a)(x-b)
_____________________=3x2-2ax-2bx-2cx+ab+ac+bc
But then I was told by a Math PhD that it is the wrong method, and from what she said, I understood that I need to do a product rule within a product rule, but the way she said it seems that I should be doing:
_________________________f '(x)= [(x-a)(x-b)][(x-c)]
Meaning, take the derivative of the first two by the product rule, and then use that as your new u', and u is just the multiplication of the first two.
I tried that and got:
__________________________________u'_______v___v'____u___________________
_______________________f '(x)= [(x-b)+(x-a)][(x-c)]+1[(x-a)(x-b)]
However, that gives me the same answer as i got before:
__________________________3x2-2ax-2bx-2cx+ab+ac+bc
My only other thought is that I should be using the derivative I got from the first two terms to do the product rule with the last term, but that would give us:
____________________________f '(x)=2(x-c)+1(2x-a-b)
________________________________=4x-a-b-2c
But that looks wrong too, because the function, if multiplied out, would give a function whose derivative is the one I obtained earlier!!!
If you can tell me where my mistake is, I'd appreciate it, thank you!!
________________________f(x)=(x-a)(x-b)(x-c).
However I am stuck as how to apply the product rule to a function like this. At first I simply let each piece be 'u' 'v' and 'w' respectively, and applied u'vw+uv'w+uvw'. Notice that the derivative of each piece is 1, so:
_________________f '(x)=1(x-b)(x-c)+(x-a)1(x-c)+(x-a)(x-b)
_____________________=3x2-2ax-2bx-2cx+ab+ac+bc
But then I was told by a Math PhD that it is the wrong method, and from what she said, I understood that I need to do a product rule within a product rule, but the way she said it seems that I should be doing:
_________________________f '(x)= [(x-a)(x-b)][(x-c)]
Meaning, take the derivative of the first two by the product rule, and then use that as your new u', and u is just the multiplication of the first two.
I tried that and got:
__________________________________u'_______v___v'____u___________________
_______________________f '(x)= [(x-b)+(x-a)][(x-c)]+1[(x-a)(x-b)]
However, that gives me the same answer as i got before:
__________________________3x2-2ax-2bx-2cx+ab+ac+bc
My only other thought is that I should be using the derivative I got from the first two terms to do the product rule with the last term, but that would give us:
____________________________f '(x)=2(x-c)+1(2x-a-b)
________________________________=4x-a-b-2c
But that looks wrong too, because the function, if multiplied out, would give a function whose derivative is the one I obtained earlier!!!
If you can tell me where my mistake is, I'd appreciate it, thank you!!