To determine

To calculate:

y'

Answer

y'= -sin⁡x.cos⁡(cos⁡x)

Explanation

Rule:

i. Chain rule:

ddxy=dydu.dudx

ii.

ddx(sin⁡x)=cos⁡x

iii.

ddx(cos⁡x)= -sin⁡x

Given:

y= sin⁡(cos⁡x)

Calculation:

y= sin⁡(cos⁡x)

Let y=sin⁡u

Where u=cos⁡x

Differentiate y with respect to x

y'=ddxsin⁡cos⁡x

By using chain rule

y'=ddu(sin⁡u).dudx

By using ddx(sin⁡x)=cos⁡x

y'=(cos⁡u).dudx

Substitute value of u

y'= cos⁡cos⁡x.ddx(cos⁡x)

By using rule ddx(cos⁡x)= -sin⁡x

y'= cos⁡cos⁡x.(-sin⁡x)

By rearrangement

y'= -sin⁡x.cos⁡(cos⁡x)

Conclusion:

y'= -sin⁡x.cos⁡(cos⁡x)