There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
Note that variables are case sensitive.\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{(3{x}^{2} + 2{({x}^{3} + 4{\frac{1}{x}}^{4})}^{\frac{1}{2}}){({x}^{2} + 2)}^{\frac{1}{2}}}{({x}^{3} - 4)}\ with\ respect\ to\ x:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{3(x^{2} + 2)^{\frac{1}{2}}x^{2}}{(x^{3} - 4)} + \frac{2(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}(x^{2} + 2)^{\frac{1}{2}}}{(x^{3} - 4)}\\&\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( \frac{3(x^{2} + 2)^{\frac{1}{2}}x^{2}}{(x^{3} - 4)} + \frac{2(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}(x^{2} + 2)^{\frac{1}{2}}}{(x^{3} - 4)}\right)}{dx}\\=&3(\frac{-(3x^{2} + 0)}{(x^{3} - 4)^{2}})(x^{2} + 2)^{\frac{1}{2}}x^{2} + \frac{3(\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 2)^{\frac{1}{2}}})x^{2}}{(x^{3} - 4)} + \frac{3(x^{2} + 2)^{\frac{1}{2}}*2x}{(x^{3} - 4)} + \frac{2(\frac{\frac{1}{2}(3x^{2} + \frac{4*-4}{x^{5}})}{(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}})(x^{2} + 2)^{\frac{1}{2}}}{(x^{3} - 4)} + 2(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}(\frac{-(3x^{2} + 0)}{(x^{3} - 4)^{2}})(x^{2} + 2)^{\frac{1}{2}} + \frac{2(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}(\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 2)^{\frac{1}{2}}})}{(x^{3} - 4)}\\=&\frac{-9(x^{2} + 2)^{\frac{1}{2}}x^{4}}{(x^{3} - 4)^{2}} + \frac{3x^{3}}{(x^{3} - 4)(x^{2} + 2)^{\frac{1}{2}}} + \frac{6(x^{2} + 2)^{\frac{1}{2}}x}{(x^{3} - 4)} + \frac{3(x^{2} + 2)^{\frac{1}{2}}x^{2}}{(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}(x^{3} - 4)} - \frac{16(x^{2} + 2)^{\frac{1}{2}}}{(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}(x^{3} - 4)x^{5}} - \frac{6(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}(x^{2} + 2)^{\frac{1}{2}}x^{2}}{(x^{3} - 4)^{2}} + \frac{2(x^{3} + \frac{4}{x^{4}})^{\frac{1}{2}}x}{(x^{2} + 2)^{\frac{1}{2}}(x^{3} - 4)}\\ \end{split}\end{equation} \]Your problem has not been solved here? Please go to the Hot Problems section!