What is the derivative of cos[sin^-1 (2w)]cos[sin1(2w)]?

1 Answer
Feb 12, 2017

dy/(dw)=-(4w)/(sqrt(1-4w^2))dydw=4w14w2

Explanation:

Let y=cos(sin^-1(2w))y=cos(sin1(2w))

Let u=sin^-1(2w)u=sin1(2w)

sinu=2wsinu=2w

(du)/(dw)cosu=2dudwcosu=2

(du)/(dw)=2/cosududw=2cosu

cos^2u+sin^2u=1cos2u+sin2u=1

cos^2u=1-sin^2u=1-(2w)^2cos2u=1sin2u=1(2w)2

cosu=sqrt(1-4w^2)cosu=14w2

(du)/(dw)=2/sqrt(1-4w^2)dudw=214w2

d/(dw)cos(u)=-sin(u)xx(du)/(dw)ddwcos(u)=sin(u)×dudw

=-sin(sin^-1(2w))xx2/(sqrt(1-4w^2))=sin(sin1(2w))×214w2

=-(4w)/(sqrt(1-4w^2))=4w14w2