There are 1 questions in this calculation: for each question, the 4 derivative of z is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {({R}^{2} + {(z + (\frac{a}{2}))}^{2})}^{\frac{-3}{2}}\ with\ respect\ to\ z：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{3}{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{3}{2}}}\right)}{dz}\\=&(\frac{\frac{-3}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}})\\=&\frac{-3z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}} - \frac{3a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}} - \frac{3a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\right)}{dz}\\=&-3(\frac{\frac{-5}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}})z - \frac{3}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}} - \frac{3(\frac{\frac{-5}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}})a}{2} + 0\\=&\frac{15z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15az}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15a^{2}}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{3}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{15z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15az}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15a^{2}}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{3}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\right)}{dz}\\=&15(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})z^{2} + \frac{15*2z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + 15(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})az + \frac{15a}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})a^{2}}{4} + 0 - 3(\frac{\frac{-5}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}})\\=&\frac{-105z^{3}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} - \frac{315az^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{315a^{2}z}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{105a^{3}}{8(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-105z^{3}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} - \frac{315az^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{315a^{2}z}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{105a^{3}}{8(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}}\right)}{dz}\\=&-105(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})z^{3} - \frac{105*3z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} - \frac{315(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})az^{2}}{2} - \frac{315a*2z}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + 45(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})z + \frac{45}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{315(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})a^{2}z}{4} - \frac{315a^{2}}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})a}{2} + 0 - \frac{105(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})a^{3}}{8} + 0\\=&\frac{945z^{4}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} + \frac{1890az^{3}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} - \frac{630z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{2835a^{2}z^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} - \frac{630az}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{945a^{3}z}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} - \frac{315a^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{945a^{4}}{16(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} + \frac{45}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}}\\ \end{split}\end{equation} \]

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