Question

find the second derivative of the function.
f(x)=(2x^4+1)^7

Answer 1

can you find the 1st derivative..?

Answer 2

yea

Answer 3

do you know chain rule of derivative

Answer 4

what is it...?

Answer 5

let me work it out.....

Answer 6

====7(2x^4+1)^6*8x

Answer 7

u forgot 8x^exponent

Answer 8

oooo 8x^3

Answer 9

if we expand by binomail theorem it can be more easy to find second derivative

Answer 10

you are missing a power but its basically correct..

\[f'(x) = 56x^3(2x^4 + 1)^6\]

do you know how to use the product rule...

\[f(x) = g(x)\times h(x)\]

where

\[g(x) = 56x^3...... and ....h(x) = (2x^4 + 1)^6\]

Answer 11

yes!

Answer 12

so does that help...?

Answer 13

yes. the only problem is that my teacher gave me the answer to be =====56x^2(1+2x^4)^5(3+54x^4)

Answer 14

will i get this if i apply the product rule?

Answer 15

ok... so you teacher has taken out common factors of 56x^2 and (2x^4 + 1)^5

Answer 16

can you try the product rule on the 1st derivative and then we'll look at the factoring

Answer 17

im not sure how to do that because just looking at the problem i would think using chain rule

Answer 18

ok...

\[g(x) = 56x^3 .....so ....g'(x) = 3 \times 56x^2\]

and

\[h(x) = (2x^4 + 1)^6 .... so.... h'(x) = 6 \times(2x^4 + 1)^5 \times 8x^3\]

does this make sense..?

Answer 19

this the chain rule agin right on the 2nd derivative?

Answer 20

then \[f''(x) = g(x)\times h'(x) + h(x) \times g'(x)\]

Answer 21

ooooooooooooo chain rule on the 2nd derivative and then product rule?

Answer 22

not sure how your teacher got 54x^4 but we'll work through it

Answer 23

ok :] no it was 56x^2

Answer 24

ok... lets see

\[f"(x) = 56x^3 \times 48x^3(2x^4 + 1)^5 + (2x^4 + 1)^6 \times 3 \times 56x^2\]

does that make sense...?

Answer 25

aand it all works out to your teacher's answer...

Answer 26

i'll try this way on my own thanks so much! but okay doing it the other way, chainrule, then chain rule then product rule the answer is still correct right?

Answer 27

yep... so I'm going to rewrite the times a little

\[56x^2 \times x \times 48x^3 \times(2x^4 + 1)^5 + (2x^4 + 1)^5 \times (2x^4 +1) \times 3 \times 56x^2\]


which shows that the common factor is

\[56x^2(2x^4 + 1)^5\]

Answer 28

this means that you will have

\[56x^2(2x^4 + 1)^5 (x \times 48x^3 + (2x^4 + 1) \times 3)\]

or

\[56x^2(2x^4 + 1)^5(48x^4 + 6x^4 + 1)\]

which finally simplifies to

\[56x^2(2x^4 + 1)^5( 54x^4 + 3)\]

you teacher's answer.

Answer 29

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Answer 30

and you need to use chain rule and product rule for the 2nd derivative.

Answer 31

opps slight typo... 2nd last line of the solution the brackets should read

\[(48x^4 + 6x^4 + 3)\]