There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{axx + bx + c}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{ax^{2} + bx + c}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{ax^{2} + bx + c}\right)}{dx}\\=&e^{ax^{2} + bx + c}(a*2x + b + 0)\\=&2axe^{ax^{2} + bx + c} + be^{ax^{2} + bx + c}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2axe^{ax^{2} + bx + c} + be^{ax^{2} + bx + c}\right)}{dx}\\=&2ae^{ax^{2} + bx + c} + 2axe^{ax^{2} + bx + c}(a*2x + b + 0) + be^{ax^{2} + bx + c}(a*2x + b + 0)\\=&2ae^{ax^{2} + bx + c} + 4a^{2}x^{2}e^{ax^{2} + bx + c} + 4abxe^{ax^{2} + bx + c} + b^{2}e^{ax^{2} + bx + c}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2ae^{ax^{2} + bx + c} + 4a^{2}x^{2}e^{ax^{2} + bx + c} + 4abxe^{ax^{2} + bx + c} + b^{2}e^{ax^{2} + bx + c}\right)}{dx}\\=&2ae^{ax^{2} + bx + c}(a*2x + b + 0) + 4a^{2}*2xe^{ax^{2} + bx + c} + 4a^{2}x^{2}e^{ax^{2} + bx + c}(a*2x + b + 0) + 4abe^{ax^{2} + bx + c} + 4abxe^{ax^{2} + bx + c}(a*2x + b + 0) + b^{2}e^{ax^{2} + bx + c}(a*2x + b + 0)\\=&12a^{2}xe^{ax^{2} + bx + c} + 6abe^{ax^{2} + bx + c} + 8a^{3}x^{3}e^{ax^{2} + bx + c} + 12a^{2}bx^{2}e^{ax^{2} + bx + c} + 6ab^{2}xe^{ax^{2} + bx + c} + b^{3}e^{ax^{2} + bx + c}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 12a^{2}xe^{ax^{2} + bx + c} + 6abe^{ax^{2} + bx + c} + 8a^{3}x^{3}e^{ax^{2} + bx + c} + 12a^{2}bx^{2}e^{ax^{2} + bx + c} + 6ab^{2}xe^{ax^{2} + bx + c} + b^{3}e^{ax^{2} + bx + c}\right)}{dx}\\=&12a^{2}e^{ax^{2} + bx + c} + 12a^{2}xe^{ax^{2} + bx + c}(a*2x + b + 0) + 6abe^{ax^{2} + bx + c}(a*2x + b + 0) + 8a^{3}*3x^{2}e^{ax^{2} + bx + c} + 8a^{3}x^{3}e^{ax^{2} + bx + c}(a*2x + b + 0) + 12a^{2}b*2xe^{ax^{2} + bx + c} + 12a^{2}bx^{2}e^{ax^{2} + bx + c}(a*2x + b + 0) + 6ab^{2}e^{ax^{2} + bx + c} + 6ab^{2}xe^{ax^{2} + bx + c}(a*2x + b + 0) + b^{3}e^{ax^{2} + bx + c}(a*2x + b + 0)\\=&12a^{2}e^{ax^{2} + bx + c} + 48a^{3}x^{2}e^{ax^{2} + bx + c} + 48a^{2}bxe^{ax^{2} + bx + c} + 12ab^{2}e^{ax^{2} + bx + c} + 16a^{4}x^{4}e^{ax^{2} + bx + c} + 32a^{3}bx^{3}e^{ax^{2} + bx + c} + 24a^{2}b^{2}x^{2}e^{ax^{2} + bx + c} + 8ab^{3}xe^{ax^{2} + bx + c} + b^{4}e^{ax^{2} + bx + c}\\ \end{split}\end{equation} \]

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