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C3 super insane - second derrivatives - any help will be really appreciated

hi everyone just got some really hard (in my opinion obviously) questions.
this is my working:

a) find d^2y/dx^2 of: SOLVED SEE BELOW!!

y= (x^3 + 2x -1)^3

dy/dx= 3(x^3 + 2x - 1)^2 (3x^2 + 2) = (9x^2 + 6)( x^3 + 2x - 1)^2

d^2y/dx^2 = [18x( x^3 +2x -1)^2] + [ 2(3x^2 + 2)(x^3 + 2x - 1)(9x^2 + 6)]

using the product rule right?

but... the ans got : 6(x^3 + 2x - 1)(12x^4 + 18x^2 - 3x + 4)

***SOLVED SEE FURTHER DOWN THREAD***









b) a bit easier.. :/ (same thing again again)

y = ln(x^2 -1)

dy/dx = 1/(x^2 - 1)

d^2y/dx^2 = -2x(x^2 - 1)^-2

they got: 2(x^2 +1)/ (x^2 - 1)^2

an extra 2 on the nominator from somewhere...cant see it.. :s








c) y= (1/(x^2 + 1))^1/2 = (x^2 + 1)^-1/2

dy/dx= -x(x^2+ 1)^-3/2

f''(x) = [-(x^2 + 1)^-3/2] + [-3x^2(x^2 + 1)^ -5/2]

but again they got: 2x^2 -1/ [(x^2 +1)^2 (x^2 + 1)^1/2]

i need serious help guys soo thanks to anyone who replies to any of these questions!!!
Reply 1
On question b) the derivate is 2x/(x^2-1) since you also have to multiply it with the inner derivate

and on c) you should put -1/2 infront of it instead of x and you should also multiply it with the inner derivate
:smile:
Reply 2
Ammar21
hi everyone just got some really hard (in my opinion obviously) questions.
this is my working:

a) find d^2y/dx^2 of:

y= (x^3 + 2x -1)^3

dy/dx= 3(x^3 + 2x - 1)^2 (3x^2 + 2) = (9x^2 + 6)( x^3 + 2x - 1)^2

d^2y/dx^2 = [18x( x^3 +2x -1)^2] + [ 2(3x^2)(x^3 + 2x - 1)(6x^2 + 4)]

using the product rule right?

but... the ans got : 6(x^3 + 2x - 1)(12x^4 + 18x^2 - 3x + 4)

soo yeah, got no clue how to factorise/simplify mine to get to theirs :frown:


Bold part (1) - I believe here you are diffrentiating the function in the brackets? If that's the case than it would be 3x^2 + 2

Bold part (2) - I'm not sure what you've done here, I think you've multiplied the function that you're taking the derivative of by 2 and diffrentiated. I think it would work if you left it as (9x^2 + 6)
Reply 3
I just done the first question, and your d^2y/dx^2 should be :

[(9x2+6)2(3x2+2)(x3+2x1)]+[(x3+2x1)2(18x)] [(9x^2 + 6)2(3x^2+2)( x^3 +2x -1)] + [ (x^3 + 2x - 1)^2(18x)]

Ill leave it there to let you figure out how to get to the final answer. (Hint, you can take the factor x^3 + 2x - 1 out.)
Reply 4
and on c) you should put -1/2 infront of it instead of x and you should also multiply it with the inner derivate
:smile:

thanks - yeha missed the minus!

it would go to:

y=(x^2 + 1)^-1/2

dy/dx = -1/2(x^2 + 1)^-3/2 * 2x = -x(x^2 +1)

right?? hopefully - coz uv still gotta multiply it by the derrivative of the brackets which is 2x
Reply 5
samir12
I just done the first question, and your d^2y/dx^2 should be :

[[color="Red"](9x2+6)[/color]2(3x2+2)(x3+2x1)]+[(x3+2x1)2(18x)] [[color="Red"](9x^2 + 6)[/color]2(3x^2+2)( x^3 +2x -1)] + [ (x^3 + 2x - 1)^2(18x)]

Ill leave it there to let you figure out how to get to the final answer. (Hint, you can take the factor x^3 + 2x - 1 out.)


thanks!!!! got lost there!!
Reply 6
thanks solved a) :

18x(x^3 + 2x -1)^2 + [(6x^2 + 2)(9x^2 + 6)(x^3 + 2x -1)

= (x^3 + 2x -1) [18x^4 + 36x^2 -18x + 56x^4 + 72x^2 + 24]

= 6(x^3 + 2x -1)(12x^4 + 18x^2 - 3x + 4)


wooop!! thanks samir12
Reply 7
inspired.
Bold part (1) - I believe here you are diffrentiating the function in the brackets? If that's the case than it would be 3x^2 + 2

Bold part (2) - I'm not sure what you've done here, I think you've multiplied the function that you're taking the derivative of by 2 and diffrentiated. I think it would work if you left it as (9x^2 + 6)


thanks!! thats put me bak on track! :smile:

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