Expert: Paul Klarreich - 2/28/2007

Question
Find 1st and 2nd Derivative,
1.) y = sin(cosx)
2.) y = tan(xsinx)

I need it now.. Thank you very much

Answer
Questioner:   Julie
Category:  Calculus
 
Subject:  Derivative of Trigonometric Functions
Question:  Find 1st and 2nd Derivative,
1.) y = sin(cosx)
2.) y = tan(x sinx)

I need it now.. Thank you very much
.......................................
Hi, Julie,

These look strange, indeed.  However, that is no excuse, so let's go:

y = sin(cos x)
Chain rule, with  y = sin u,  u = cos x

dy/du = cos u
du/dx = - sin x

dy   dy du
-- = -- --
dx   du dx

= cos u (- sin x)
= - sin x cos(cos x)
.....................................

y = tan(x sinx)

Same stuff, but now you need the product rule for the 'inner' derivative.

y = tan u,  u = x sin x


du/dx = (x)(cos x) + (1)(sin x) = x cos x + sin x


dy/du = sec^2(u)

dy   dy du
-- = -- --
dx   du dx

= sec^2(u)(x cos x + sin x)
= (x cos x + sin x)sec^2(x sin x)