Mister Exam

Derivative of (a*arctg(x))/(a+ln(x))

Function f() - derivative -N order at the point
v

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The solution

You have entered [src]
a*atan(x) 
----------
a + log(x)
$$\frac{a \operatorname{atan}{\left(x \right)}}{a + \log{\left(x \right)}}$$
(a*atan(x))/(a + log(x))
The first derivative [src]
          a                a*atan(x)   
--------------------- - ---------------
/     2\                              2
\1 + x /*(a + log(x))   x*(a + log(x)) 
$$\frac{a}{\left(a + \log{\left(x \right)}\right) \left(x^{2} + 1\right)} - \frac{a \operatorname{atan}{\left(x \right)}}{x \left(a + \log{\left(x \right)}\right)^{2}}$$
The second derivative [src]
  /                                        /        2     \        \
  |                                        |1 + ----------|*atan(x)|
  |     2*x                 2              \    a + log(x)/        |
a*|- --------- - ----------------------- + ------------------------|
  |          2     /     2\                     2                  |
  |  /     2\    x*\1 + x /*(a + log(x))       x *(a + log(x))     |
  \  \1 + x /                                                      /
--------------------------------------------------------------------
                             a + log(x)                             
$$\frac{a \left(- \frac{2 x}{\left(x^{2} + 1\right)^{2}} - \frac{2}{x \left(a + \log{\left(x \right)}\right) \left(x^{2} + 1\right)} + \frac{\left(1 + \frac{2}{a + \log{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)}}{x^{2} \left(a + \log{\left(x \right)}\right)}\right)}{a + \log{\left(x \right)}}$$
The third derivative [src]
  /  /         2 \                                                                                                 \
  |  |      4*x  |                              /        3              3      \                                   |
  |2*|-1 + ------|                            2*|1 + ---------- + -------------|*atan(x)        /        2     \   |
  |  |          2|                              |    a + log(x)               2|              3*|1 + ----------|   |
  |  \     1 + x /             6                \                 (a + log(x)) /                \    a + log(x)/   |
a*|--------------- + ---------------------- - ------------------------------------------ + ------------------------|
  |           2              2                              3                               2 /     2\             |
  |   /     2\       /     2\                              x *(a + log(x))                 x *\1 + x /*(a + log(x))|
  \   \1 + x /       \1 + x / *(a + log(x))                                                                        /
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                                                     a + log(x)                                                     
$$\frac{a \left(\frac{2 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{6}{\left(a + \log{\left(x \right)}\right) \left(x^{2} + 1\right)^{2}} + \frac{3 \left(1 + \frac{2}{a + \log{\left(x \right)}}\right)}{x^{2} \left(a + \log{\left(x \right)}\right) \left(x^{2} + 1\right)} - \frac{2 \left(1 + \frac{3}{a + \log{\left(x \right)}} + \frac{3}{\left(a + \log{\left(x \right)}\right)^{2}}\right) \operatorname{atan}{\left(x \right)}}{x^{3} \left(a + \log{\left(x \right)}\right)}\right)}{a + \log{\left(x \right)}}$$