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