Homework Help

[vyvance]

I've been stuck on this problem for a while now and just can't seem to figure it out. If anyone knows how to do the problem, the help would be much appreciated. I know the answer for the problem, I just can't figure out how to get there.

Problem:
Find f'(x), f''(x), and f'''(x).
f(x)= (x^2 - 3x)/(4x+2)

I found f'(x) using quotient rule: f'(x)=(4x^2 + 4x -6)/(4x+2)^2

I can't figure out how to solve for the second two derivatives though. The answers for the subsequent derivatives of f(x) are:
f''(x)= 56/(4x+2)^3
f'''(x)= -672/(4x+2)^4

I assume that since the exponent of the denominator is increasing by one each time, that the extended power rule is being used somehow. Beyond that I don't know. Do I use quotient rule again for the next two derivatives or extended power rule or what? I can't seem to get the math to work.

If anyone knows the steps to reach the answer, the help is much appreciated. Thanks in advance if you do. Only problem I haven't been able to solve so far, and it is bugging me.

[Sunandmoon]

Hi there,

You can re-write f'(x) like so:

f'(x) = (4x^2 + 4x - 6)(4x + 2)^(-2)

Then using the product rule (derivative of f*g = f'*g + f*g'), you get

f''(x) = (8x + 4)(4x + 2)^(-2) + (4x^2 + 4x - 6)(-2)(4x + 2)^(-3)(4)

The second term (4x^2 + 4x - 6)(-2)(4x + 2)^(-3)(4) might look weird because we needed to take the derivative of (4x + 2)^(-2). So bring down the exponent (-2), multiply it by the original term with the exponent reduced by one, and multiply it by the derivative of the inside.

f''(x) = ( (8x + 4) / (4x + 2)^2 ) + ( ( -8 * (4x^2 + 4x - 6) ) / (4x + 2)^3 )
f''(x) = ( (8x + 4)(4x + 2) - 8(4x^2 + 4x - 6) ) / (4x + 2)^3

Note we make the exponent positive for the denominator since it's being divided by again.

f''(x) = ( 32x^2 + 16x + 16x + 8 - 32x^2 - 32x + 48 ) / (4x + 2)^3
f''(x) = 56 / (4x + 2)^3

f'''(x) = 56 * (4x + 2)^(-3)
f'''(x) = 0 + (56)(-3)(4x + 2)^(-4)(4)
f'''(x) = -672 / (4x + 2)^4

Hope that helps!

[firemaker]

What was ^ stand for

[Sunandmoon]

Originally Posted by firemaker

What was ^ stand for

Exponent

[kellymporta]

Just one word to solve all your problems: wolframalpha.com

[vyvance]

Thanks a bunch. I was missing the derivative of the denominator when bringing it up.Think I was bringing the exponent on the denominator to -3 before trying to solve for the derivative or something. Or rather, putting -3 out in front instead of -2.

I had done multiple derivatives of a function before in the homework, but it had been something simple like f(x)=1/x^3.

Thanks again, you saved me. Now time to do some derivatives of trigonometric functions. Seems pretty simple so far, but then so did this last assignment till this problem lol.

[midwestgirl89]

I think my brain just exploded a bit looking at that problem. You are awesome to be smart enough to even know what that means.

I'm a social science person so that equation makes me feel queesy. Good luck on the rest of your hw. : )

[TheRoof]

ahhh calculus....
I'm sure I could have helped you if you asked this this past spring because I took calc last semester. Unfortunately my brain seemed to have erased all memory of calc and derivatives and shit

[Chip]

Originally Posted by midwestgirl89

I think my brain just exploded a bit looking at that problem. You are awesome to be smart enough to even know what that means.

I'm a social science person so that equation makes me feel queesy. Good luck on the rest of your hw. : )

Totally agree. Social sciences, written language, all that sort of thing I'm fine with... anything above basic mathematics, particularly chicken scratches like the above, not so much

[Gerry]

I'm glad others have already helped you with your homework because math was always my worst subject and looking at the step-by-step process sunandmoon did wants to give me a headache.