There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ a{(x + y + z)}^{4} + b{(x + y + z)}^{3} + c{(x + y + z)}^{2} + d(x + y + z) + f\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ax^{4} + 4ayx^{3} + 4azx^{3} + 6ay^{2}x^{2} + 12ayzx^{2} + 6az^{2}x^{2} + 4ay^{3}x + 12ay^{2}zx + 12ayz^{2}x + 4az^{3}x + 4ay^{3}z + 6ay^{2}z^{2} + 4ayz^{3} + ay^{4} + az^{4} + bx^{3} + 3ybx^{2} + 3zbx^{2} + 3y^{2}bx + 6yzbx + 3z^{2}bx + y^{3}b + 3y^{2}zb + 3yz^{2}b + z^{3}b + cx^{2} + 2ycx + 2zcx + y^{2}c + 2yzc + z^{2}c + dx + yd + zd + f\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ax^{4} + 4ayx^{3} + 4azx^{3} + 6ay^{2}x^{2} + 12ayzx^{2} + 6az^{2}x^{2} + 4ay^{3}x + 12ay^{2}zx + 12ayz^{2}x + 4az^{3}x + 4ay^{3}z + 6ay^{2}z^{2} + 4ayz^{3} + ay^{4} + az^{4} + bx^{3} + 3ybx^{2} + 3zbx^{2} + 3y^{2}bx + 6yzbx + 3z^{2}bx + y^{3}b + 3y^{2}zb + 3yz^{2}b + z^{3}b + cx^{2} + 2ycx + 2zcx + y^{2}c + 2yzc + z^{2}c + dx + yd + zd + f\right)}{dx}\\=&a*4x^{3} + 4ay*3x^{2} + 4az*3x^{2} + 6ay^{2}*2x + 12ayz*2x + 6az^{2}*2x + 4ay^{3} + 12ay^{2}z + 12ayz^{2} + 4az^{3} + 0 + 0 + 0 + 0 + 0 + b*3x^{2} + 3yb*2x + 3zb*2x + 3y^{2}b + 6yzb + 3z^{2}b + 0 + 0 + 0 + 0 + c*2x + 2yc + 2zc + 0 + 0 + 0 + d + 0 + 0 + 0\\=&4ax^{3} + 12ayx^{2} + 12azx^{2} + 12ay^{2}x + 24ayzx + 12az^{2}x + 12ay^{2}z + 12ayz^{2} + 4ay^{3} + 4az^{3} + 3bx^{2} + 6ybx + 6zbx + 3y^{2}b + 6yzb + 3z^{2}b + 2cx + 2yc + 2zc + d\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 4ax^{3} + 12ayx^{2} + 12azx^{2} + 12ay^{2}x + 24ayzx + 12az^{2}x + 12ay^{2}z + 12ayz^{2} + 4ay^{3} + 4az^{3} + 3bx^{2} + 6ybx + 6zbx + 3y^{2}b + 6yzb + 3z^{2}b + 2cx + 2yc + 2zc + d\right)}{dx}\\=&4a*3x^{2} + 12ay*2x + 12az*2x + 12ay^{2} + 24ayz + 12az^{2} + 0 + 0 + 0 + 0 + 3b*2x + 6yb + 6zb + 0 + 0 + 0 + 2c + 0 + 0 + 0\\=&12ax^{2} + 24ayx + 24azx + 24ayz + 12ay^{2} + 12az^{2} + 6bx + 6yb + 6zb + 2c\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 12ax^{2} + 24ayx + 24azx + 24ayz + 12ay^{2} + 12az^{2} + 6bx + 6yb + 6zb + 2c\right)}{dx}\\=&12a*2x + 24ay + 24az + 0 + 0 + 0 + 6b + 0 + 0 + 0\\=&24ax + 24ay + 24az + 6b\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 24ax + 24ay + 24az + 6b\right)}{dx}\\=&24a + 0 + 0 + 0\\=&24a\\ \end{split}\end{equation} \]

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