Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
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Now simplify:
The answer is:
The first derivative
[src]
___
\/ x / / 2 \ 3/2\
/ 2 \ |log\x + 1/ 2*x |
\x + 1/ *|----------- + ------|
| ___ 2 |
\ 2*\/ x x + 1/
$$\left(x^{2} + 1\right)^{\sqrt{x}} \left(\frac{2 x^{\frac{3}{2}}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{2 \sqrt{x}}\right)$$
The second derivative
[src]
/ 2 \
|/ / 2\ 3/2\ |
||log\1 + x / 4*x | |
___ ||----------- + ------| |
\/ x || ___ 2| 5/2 ___ / 2\|
/ 2\ |\ \/ x 1 + x / 4*x 4*\/ x log\1 + x /|
\1 + x / *|----------------------- - --------- + ------- - -----------|
| 4 2 2 3/2 |
| / 2\ 1 + x 4*x |
\ \1 + x / /
$$\left(x^{2} + 1\right)^{\sqrt{x}} \left(- \frac{4 x^{\frac{5}{2}}}{\left(x^{2} + 1\right)^{2}} + \frac{4 \sqrt{x}}{x^{2} + 1} + \frac{\left(\frac{4 x^{\frac{3}{2}}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{\sqrt{x}}\right)^{2}}{4} - \frac{\log{\left(x^{2} + 1 \right)}}{4 x^{\frac{3}{2}}}\right)$$
The third derivative
[src]
/ 3 / / 2\ 3/2\ / / 2\ ___ 5/2 \ \
|/ / 2\ 3/2\ |log\1 + x / 4*x | |log\1 + x / 16*\/ x 16*x | |
||log\1 + x / 4*x | 3*|----------- + ------|*|----------- - -------- + ---------| |
___ ||----------- + ------| | ___ 2| | 3/2 2 2| |
\/ x || ___ 2| 3/2 7/2 \ \/ x 1 + x / | x 1 + x / 2\ | / 2\|
/ 2\ |\ \/ x 1 + x / 18*x 16*x \ \1 + x / / 3 3*log\1 + x /|
\1 + x / *|----------------------- - --------- + --------- - ------------------------------------------------------------- + ---------------- + -------------|
| 8 2 3 8 ___ / 2\ 5/2 |
| / 2\ / 2\ 2*\/ x *\1 + x / 8*x |
\ \1 + x / \1 + x / /
$$\left(x^{2} + 1\right)^{\sqrt{x}} \left(\frac{16 x^{\frac{7}{2}}}{\left(x^{2} + 1\right)^{3}} - \frac{18 x^{\frac{3}{2}}}{\left(x^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{\frac{3}{2}}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{\sqrt{x}}\right)^{3}}{8} - \frac{3 \left(\frac{4 x^{\frac{3}{2}}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{\sqrt{x}}\right) \left(\frac{16 x^{\frac{5}{2}}}{\left(x^{2} + 1\right)^{2}} - \frac{16 \sqrt{x}}{x^{2} + 1} + \frac{\log{\left(x^{2} + 1 \right)}}{x^{\frac{3}{2}}}\right)}{8} + \frac{3}{2 \sqrt{x} \left(x^{2} + 1\right)} + \frac{3 \log{\left(x^{2} + 1 \right)}}{8 x^{\frac{5}{2}}}\right)$$