#frac{dy}{dx} = frac{ frac{dy}{d theta} }{ frac{dx}{d theta} }#
#= frac{ frac{d}{d theta}(rsintheta) }{ frac{d}{d theta}(rcostheta) }#
#= frac{ rcostheta + frac{dr}{d theta}sintheta }{ - rsintheta + frac{dr}{d theta}costheta }#
#frac{dr}{d theta} = frac{d}{d theta}(2theta + 3cos(theta/2 - frac{4pi}{3}))#
#= 2 - 3/2 sin(theta/2 - frac{4pi}{3})#
#frac{dr}{d theta}_{|theta=frac{5pi}{4}} = 2 - 3/2 sin((frac{5pi}{4})/2 - frac{4pi}{3})#
#= 2 - 3/2 sin(frac{-17pi}{24})#
#= 2 + 3/2 sin(frac{7pi}{24})#
#frac{dy}{dx}_{|theta=frac{5pi}{4}} = frac{ (2(frac{5pi}{4}) + 3cos(frac{5pi}{4}/2 - frac{4pi}{3}))cosfrac{5pi}{4} + frac{dr}{d theta}\_{|theta=frac{5pi}{4}} sinfrac{5pi}{4} }{ - (2(frac{5pi}{4}) + 3cos(frac{5pi}{4}/2 - frac{4pi}{3}))sinfrac{5pi}{4} + frac{dr}{d theta}\_{|theta=frac{5pi}{4}} cosfrac{5pi}{4} }#
#= frac{ (frac{5pi}{2} + 3cos(frac{-17pi}{24}))(-1/sqrt{2}) + frac{dr}{d theta}\_{|theta=frac{5pi}{4}} (-1/sqrt{2}) }{ - (frac{5pi}{2} + 3cos(frac{-17pi}{24}))(-1/sqrt{2}) + frac{dr}{d theta}\_{|theta=frac{5pi}{4}} (-1/sqrt{2}) }#
#= frac{ frac{5pi}{2} + 3cos(frac{17pi}{24}) + frac{dr}{d theta}\_{|theta=frac{5pi}{4}} }{ - frac{5pi}{2} - 3cos(frac{17pi}{24}) + frac{dr}{d theta}\_{|theta=frac{5pi}{4}} }#
#= frac{ frac{5pi}{2} + 3cos(frac{17pi}{24}) + (2 + 3/2 sin(frac{7pi}{24})) }{ - frac{5pi}{2} - 3cos(frac{17pi}{24}) + (2 + 3/2 sin(frac{7pi}{24})) }#
#= frac{ 4 + 5pi - 6cos(frac{7pi}{24}) + 3 sin(frac{7pi}{24}) }{ 4 - 5pi + 6cos(frac{7pi}{24}) + 3 sin(frac{7pi}{24}) }#
#~~ -3.24835#
