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

1. satellite73

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

2. onegirl

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

3. onegirl

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

4. onegirl

@satellite73 ?

5. onegirl

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

which one : |dw:1362427439289:dw|

7. onegirl

cosx^2

so, the first one in my diagram ?

9. onegirl

yes

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

11. onegirl

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

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

13. onegirl

yes

14. onegirl

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

yes

16. onegirl

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?

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 ' = ... ?

18. onegirl

how would i do it?

19. onegirl

can u show it?

do u know derivative of -2x ?

21. onegirl

yea

22. onegirl

its -2 right?

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

24. onegirl

okay hold on

25. onegirl

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

26. onegirl

for my derivative of cosx^2

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

28. onegirl

okay

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

30. onegirl

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

31. onegirl

and i got cos (x) sin(x)

32. onegirl

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

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

sorry, it should v' = 2xcosx^2

35. onegirl

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

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

37. onegirl

ok

38. onegirl

so what do i do next?

nothing, it is clear :)

40. onegirl

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

41. onegirl

i meant f''(x)

42. onegirl

are you there?

43. onegirl