pls show how is differentiate w.r.t t (sin(xt+ylnt))^2*(sqrt(xt+ylnt))?

Expert Answers
sciencesolve eNotes educator| Certified Educator

Since the problem specifies to differentiate with respect to variable t, hence, you should consider x and y as constants.

You need to use product and differentiation rules such that:

`f'(t) = ((sin(xt+ylnt))^2)'*(sqrt(xt+ylnt)) + ((sin(xt+ylnt))^2)*(sqrt(xt+ylnt))'`

`f'(t) = 2sin(xt+ylnt)*cos(xt+ylnt)*(x + y/t)*(sqrt(xt+ylnt)) + ((sin(xt+ylnt))^2)*(1/(2sqrt(xt+ylnt))*(x + y/t))`

You should factor out `(x + y/t)`  and you need to convert the product `2sin(xt+ylnt)*cos(xt+ylnt)`  into the sine of double angle such that:

`f'(t) = (x + y/t)*(sin 2(xt+ylnt)*sqrt(xt+ylnt) + ((sin(xt+ylnt))^2)/(2sqrt(xt+ylnt)))`

Hence, differentiating with respect to t yields `f'(t) = (x + y/t)*(sin 2(xt+ylnt)*sqrt(xt+ylnt) + ((sin(xt+ylnt))^2)/(2sqrt(xt+ylnt))).`