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 $h^2(x) = x^2sin^2(x) + 2xsin(x)cos(x) + cos^2(x)$

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

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

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

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

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