Question #183ce

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Answer 1 · 2017-01-14T17:22:40

Given: #f(x)=(sin(x)-xcos(x))/(xsin(x)+cos(x))#

#f(x)# is of the form #g(x)/(h(x))#, therefore, The Quotient Rule applies.

let #g(x) = sin(x) - xcos(x)#

then #g'(x) = cos(x) - cos(x) + xsin(x) = xsin(x)#

let #h(x) = xsin(x) + cos(x)#

then #h'(x) = sin(x) + xcos(x) - sin(x) = xcos(x)#

and #h^2(x) = x^2sin^2(x) + 2xsin(x)cos(x) + cos^2(x)#

#f'(x) = (g'(x)h(x) - g(x)h'(x))/(h^2(x))#

#f'(x) = ((xsin(x))(xsin(x) + cos(x)) - (sin(x) - xcos(x))(xcos(x)))/(x^2sin^2(x) + 2xsin(x)cos(x) + cos^2(x))#

#f'(x) = (x^2sin^2(x) + xsin(x)cos(x) - xsin(x)cos(x) + x^2cos^2(x))/(x^2sin^2(x) + 2xsin(x)cos(x) + cos^2(x))#

#f'(x) = (x^2(sin^2(x) + cos^2(x)))/(x^2sin^2(x) + 2xsin(x)cos(x) + cos^2(x))#

#f'(x) = (x^2)/(x^2sin^2(x) + 2xsin(x)cos(x) + cos^2(x))#