Question Number 103894 by Study last updated on 18/Jul/20

(d/(d((d/dx)sinx)))∙sinx=?

Answered by 1549442205 last updated on 18/Jul/20

Denote I=(d/(d(d/dx)sinx))sinx=(d/(dcosx))sinx.  Putting  cos x=t.Then  we have I=(d/dt)(√(1−t^2 ))=((−t)/(√(1−t^2 )))=((−cosx)/(sinx))  =−cot x

Answered by MAB last updated on 18/Jul/20

  (d/(d((d/dx)sinx)))∙sinx=(d/(dcos(x)))sin(x)  =((dsin(x))/dx)∙(dx/(dcos(x)))  =cos(x)∙(−(1/(sin(x))))  =−cot(x)

Commented byStudy last updated on 18/Jul/20

how we can calculate (dx/(dcos(x)))?

Commented bybobhans last updated on 18/Jul/20

(dx/(d(cos x))) = (1/((d(cos x))/dx)) = (1/(−sin x))

Answered by bobhans last updated on 18/Jul/20

((d(sin x))/(d(((d(sin x))/dx)))) = ((d(sin x))/(d(cos x))) = ((d(sin x))/dx) × (dx/(d(cos x)))  = cos x × (1/((d(cos x))/dx)) = cos x ×(1/(−sin x))  = −cot x