onegirl
Find the indicated derivative of the function f(x) = cosx^2, find f''(x)



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anonymous
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\[2\cos(x)\sin(x)\]is your first derivative
take the derivative of that one, using the product rule

onegirl
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@satellite73 it will be 2cos(x)sin(x)

onegirl
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@genius12 so the derivative of 2cos(x) sin(x) , and i got 2 cos(2x)

onegirl
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@satellite73 ?

onegirl
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@jazy should i now take the derivative of 2 cos(2x) ? or

RadEn
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which one :
dw:1362427439289:dw

onegirl
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cosx^2

RadEn
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so, the first one in my diagram ?

onegirl
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yes

RadEn
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oke, to get the first derivative of f(x) we can use the cain rule
hint : derivative of x^2 = 2x
derivative of cos =  sin

onegirl
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yea i got it the first derivative will be 2 cos(x) sin (x) right?

RadEn
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so, derivative of f(x) = cosx^2 > f ' (x) = 2x(sinx^2) = 2x sinx^2

onegirl
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yes

onegirl
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so then i take the derivative of 2x sinx^2 ?

RadEn
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yes

onegirl
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okay after i take the derivative of 2x sinx i get 2 cos(2x) so then i take the derivative of 2 cos(2x) right?

RadEn
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use the product rule to get the 2nd derivative :
use the formula : f '' = u'v + uv'
assuming that u=2x > u' = ... ?
v = sinx^2 > v ' = ... ?

onegirl
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how would i do it?

onegirl
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can u show it?

RadEn
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do u know derivative of 2x ?

onegirl
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yea

onegirl
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its 2 right?

RadEn
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yes, that's right
also, sinx^2 ?

onegirl
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okay hold on

onegirl
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but where does sinx^2 come from i got 2 cos(x) sin(x) ??

onegirl
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for my derivative of cosx^2

RadEn
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f(x) = cosx^2 > f ' (x) = 2x(sinx^2) = 2x sinx^2

onegirl
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okay

RadEn
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it just the first derivative of f(x), now we have to deriv again to get f "
then, use the product rule

onegirl
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so now i need to find the derivative of sin^x right? since we found 2x which we got 2

onegirl
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and i got cos (x) sin(x)

onegirl
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so it will be 2 cos(x) sin(x) right?

RadEn
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2x sinx^2
let u=2x > u' = 2
v = sinx^2 > v ' = 2xcoscx^2

RadEn
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sorry, it should v' = 2xcosx^2

onegirl
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okay so how would i set it up in the formula ?

RadEn
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f " = u'v + uv'
f " = (2)sinx^2 + (2x)2xcosx^2 = 2sinx^2  4x^2cosx^2

onegirl
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ok

onegirl
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so what do i do next?

RadEn
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nothing, it is clear :)

onegirl
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okay so the f'''(x) will be 2sinx^2 = 4x^2 cosx^2 ?

onegirl
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i meant f''(x)

onegirl
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are you there?

onegirl
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@RadEn so thats the final answer?