Question

Think my textbook may have the wrong answer for this:
Find all higher derivatives -> f(r) = r(4r + 9)^3
I've tried a few different things, none are giving me the textbook answer. Please show explanation and not just the answer.

Answer 1

I see you typing dumbcow. You're awesome :)

Answer 2

this is combination of product rule and chain rule
let u = r, v = (4r+9)^3
u' = 1
v' = 4*3(4r+9)^2 = 12(4r+9)^2

f'(r) = u'v + uv'
f'(r) = (4r+9)^3 + 12r(4r+9)^2

------------------------
for 2nd derivative
let g(r) = 12r(4r+9)^2, we'll find g' first
u = 12r, v = (4r+9)^2
u' = 12
v' = 4*2(4r+9) = 8(4r+9)

g'(r) = u'v + uv'
g'(r) = 12(4r+9)^2 + 96r(4r+9)

--> f''(r) = 4*3(4r+9)^2 + 12(4r+9)^2 + 96r(4r+9)
f''(r) = 24(4r+9)^2 + 384r^2 + 864r
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for 3rd derivative,

f'''(r) = 48(4r+9) + 768r + 864

-----------------------------------
for 4th derivative

f''''(r) = 192 + 768 = 960

Answer 3

depending on how the textbook simplifies answers, it might be same thing but look different
is everything expanded out in their answers?

Answer 4

Here's the first two:
\[f \prime(r) = (16r+9)(4r+9)^{2}, f ^{n}(r) = 24(8r+9)(4r+9)\]
It's strange because I went ahead and did the question after it and had no problems, although it is a slightly different question (4 sqrt(8x-3))

Answer 5

Yeah I'm getting the same answer as you are, I'll leave a note asking my teacher to check it over himself before marking it.

Answer 6

ok answers match, just checked
the books are factored a little more nicely, thats all

Answer 7

Okay thanks. I'm stuck in between 2nd and 3rd derivative, where you do f''(r) = 4*3(4r+9)^2 + 12(4r+9)^2 + 96r(4r+9), where is that first term coming from?

Answer 8

for example:
(4r+9)^3 + 12r(4r+9)^2
factor out a (4r+9)^2
(4r+9)^2(4r+9 +12r) =(16r+9) (4r+9)^2

Answer 9

oh right sorry i was afraid of that, it comes from derivative of first term of f'(r)
-->(4r+9)^3 + 12r(4r+9)^2
^

Answer 10

Hmm, I see how you got the first f"(r) of the second derivative, but not how you got the 24(4r+9)^2 in the second step. I'll continue this from school, in about 1:30 hours. Wish me luck!

Answer 11

Ahh ...i just combined terms, 4*3=12 ...12+12 =24

good luck

Answer 12

oh and then i expanded 96r(4r+9)