There are 1 questions in this calculation: for each question, the 1 derivative of H is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ (\frac{(bH - \frac{1}{2}(c({e}^{2}))(1 - {k}^{2}))(\frac{((1 - k)e^{S} - rH)}{((1 - k)(e^{S} - rH))})}{((1 - k)(bH - \frac{1}{2}(c({e}^{2}))(1 + k)))})\ with\ respect\ to\ H：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{-bkHe^{S}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{bHe^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} - \frac{brH^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}ck^{3}e^{S}e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}ck^{2}e^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} - \frac{\frac{1}{2}ck^{2}rHe^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}cke^{S}e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}ce^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} + \frac{\frac{1}{2}crHe^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{-bkHe^{S}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{bHe^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} - \frac{brH^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}ck^{3}e^{S}e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}ck^{2}e^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} - \frac{\frac{1}{2}ck^{2}rHe^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}cke^{S}e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}ce^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} + \frac{\frac{1}{2}crHe^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)}\right)}{dH}\\=&\frac{-(\frac{-2(0 + 0)}{(-k + 1)^{3}})bkHe^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})bkHe^{S}}{(-k + 1)^{2}(e^{S} - rH)} - \frac{(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})bkHe^{S}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{bke^{S}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{bkHe^{S}*0}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})bHe^{S}}{(-k + 1)^{2}(e^{S} - rH)} + \frac{(\frac{-2(0 + 0)}{(-k + 1)^{3}})bHe^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})bHe^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}} + \frac{be^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} + \frac{bHe^{S}*0}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} - \frac{(\frac{-2(0 + 0)}{(-k + 1)^{3}})brH^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})brH^{2}}{(-k + 1)^{2}(e^{S} - rH)} - \frac{(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})brH^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{br*2H}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-2(0 + 0)}{(-k + 1)^{3}})ck^{3}e^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})ck^{3}e^{S}e^{2}}{(-k + 1)^{2}(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})ck^{3}e^{S}e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{\frac{1}{2}ck^{3}e^{S}*0e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}ck^{3}e^{S}*2e*0}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})ck^{2}e^{S}e^{2}}{(-k + 1)^{2}(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-2(0 + 0)}{(-k + 1)^{3}})ck^{2}e^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})ck^{2}e^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}} + \frac{\frac{1}{2}ck^{2}e^{S}*0e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} + \frac{\frac{1}{2}ck^{2}e^{S}*2e*0}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-2(0 + 0)}{(-k + 1)^{3}})ck^{2}rHe^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})ck^{2}rHe^{2}}{(-k + 1)^{2}(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})ck^{2}rHe^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{\frac{1}{2}ck^{2}re^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}ck^{2}rH*2e*0}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-2(0 + 0)}{(-k + 1)^{3}})cke^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})cke^{S}e^{2}}{(-k + 1)^{2}(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})cke^{S}e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} + \frac{\frac{1}{2}cke^{S}*0e^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}cke^{S}*2e*0}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})ce^{S}e^{2}}{(-k + 1)^{2}(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-2(0 + 0)}{(-k + 1)^{3}})ce^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{\frac{1}{2}(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})ce^{S}e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}} - \frac{\frac{1}{2}ce^{S}*0e^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} - \frac{\frac{1}{2}ce^{S}*2e*0}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-2(0 + 0)}{(-k + 1)^{3}})crHe^{2}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-(b - \frac{1}{2}ck*2e*0 - \frac{1}{2}c*2e*0)}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}})crHe^{2}}{(-k + 1)^{2}(e^{S} - rH)} + \frac{\frac{1}{2}(\frac{-(e^{S}*0 - r)}{(e^{S} - rH)^{2}})crHe^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} + \frac{\frac{1}{2}cre^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{\frac{1}{2}crH*2e*0}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)}\\=&\frac{b^{2}kHe^{S}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(e^{S} - rH)} + \frac{bkrHe^{S}}{(-k + 1)^{2}(e^{S} - rH)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{bke^{S}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} - \frac{b^{2}He^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(-k + 1)^{2}(e^{S} - rH)} + \frac{brHe^{S}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)^{2}} + \frac{be^{S}}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(-k + 1)^{2}(e^{S} - rH)} + \frac{b^{2}rH^{2}}{(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(e^{S} - rH)} - \frac{br^{2}H^{2}}{(-k + 1)^{2}(e^{S} - rH)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{2brH}{(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)(-k + 1)^{2}} + \frac{bck^{3}e^{S}e^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(e^{S} - rH)} - \frac{ck^{3}re^{S}e^{2}}{2(-k + 1)^{2}(e^{S} - rH)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{bck^{2}e^{S}e^{2}}{2(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(-k + 1)^{2}(e^{S} - rH)} + \frac{ck^{2}re^{S}e^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)^{2}} + \frac{bck^{2}rHe^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(e^{S} - rH)} - \frac{ck^{2}r^{2}He^{2}}{2(-k + 1)^{2}(e^{S} - rH)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} + \frac{ckre^{S}e^{2}}{2(-k + 1)^{2}(e^{S} - rH)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} - \frac{bcke^{S}e^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(e^{S} - rH)} - \frac{ck^{2}re^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)} + \frac{bce^{S}e^{2}}{2(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(-k + 1)^{2}(e^{S} - rH)} - \frac{cre^{S}e^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)^{2}} - \frac{bcrHe^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})^{2}(e^{S} - rH)} + \frac{cr^{2}He^{2}}{2(-k + 1)^{2}(e^{S} - rH)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})} + \frac{cre^{2}}{2(-k + 1)^{2}(bH - \frac{1}{2}cke^{2} - \frac{1}{2}ce^{2})(e^{S} - rH)}\\ \end{split}\end{equation} \]

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