There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ (\frac{1}{2}){a}^{2}xsqrt({x}^{2} + {a}^{2}) + (\frac{1}{2}){a}^{2}ln(sqrt({x}^{2} + {a}^{2}) - x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{2}a^{2}xsqrt(x^{2} + a^{2}) + \frac{1}{2}a^{2}ln(sqrt(x^{2} + a^{2}) - x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{2}a^{2}xsqrt(x^{2} + a^{2}) + \frac{1}{2}a^{2}ln(sqrt(x^{2} + a^{2}) - x)\right)}{dx}\\=&\frac{1}{2}a^{2}sqrt(x^{2} + a^{2}) + \frac{\frac{1}{2}a^{2}x(2x + 0)*\frac{1}{2}}{(x^{2} + a^{2})^{\frac{1}{2}}} + \frac{\frac{1}{2}a^{2}(\frac{(2x + 0)*\frac{1}{2}}{(x^{2} + a^{2})^{\frac{1}{2}}} - 1)}{(sqrt(x^{2} + a^{2}) - x)}\\=&\frac{a^{2}sqrt(x^{2} + a^{2})}{2} + \frac{a^{2}x^{2}}{2(x^{2} + a^{2})^{\frac{1}{2}}} + \frac{a^{2}x}{2(sqrt(x^{2} + a^{2}) - x)(x^{2} + a^{2})^{\frac{1}{2}}} - \frac{a^{2}}{2(sqrt(x^{2} + a^{2}) - x)}\\ \end{split}\end{equation} \]

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