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The derivative of the function/ 5x^atan(x)

Derivative of 5x^atan(x)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   atan(x)
5*x
5xatan⁡(x)5 x^{\operatorname{atan}{\left(x \right)}} 
5*x^atan(x)
Detail solution

  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Don't know the steps in finding this derivative.

      But the derivative is

      (log⁡(atan⁡(x))+1)atan⁡atan⁡(x)(x)\left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}

    So, the result is: 5(log⁡(atan⁡(x))+1)atan⁡atan⁡(x)(x)5 \left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}

The answer is:

5(log⁡(atan⁡(x))+1)atan⁡atan⁡(x)(x)5 \left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}

The graph
02468-8-6-4-2-1010-200200
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   atan(x) /atan(x)   log(x)\
5*x       *|------- + ------|
           |   x           2|
           \          1 + x /
5xatan⁡(x)(log⁡(x)x2+1+atan⁡(x)x)5 x^{\operatorname{atan}{\left(x \right)}} \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)
Simplify
The second derivative [src] 
            /                    2                                    \
    atan(x) |  /atan(x)   log(x)\    atan(x)       2        2*x*log(x)|
-5*x       *|- |------- + ------|  + ------- - ---------- + ----------|
            |  |   x           2|        2       /     2\           2 |
            |  \          1 + x /       x      x*\1 + x /   /     2\  |
            \                                               \1 + x /  /
−5xatan⁡(x)(2xlog⁡(x)(x2+1)2−(log⁡(x)x2+1+atan⁡(x)x)2−2x(x2+1)+atan⁡(x)x2)- 5 x^{\operatorname{atan}{\left(x \right)}} \left(\frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)^{2} - \frac{2}{x \left(x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}\right)
Simplify
The third derivative [src] 
            /                    3                                                                                                                   2       \
    atan(x) |  /atan(x)   log(x)\        6       2*atan(x)    2*log(x)        3          /atan(x)   log(x)\ /atan(x)       2        2*x*log(x)\   8*x *log(x)|
-5*x       *|- |------- + ------|  + --------- - --------- + --------- + ----------- + 3*|------- + ------|*|------- - ---------- + ----------| - -----------|
            |  |   x           2|            2        3              2    2 /     2\     |   x           2| |    2       /     2\           2 |            3 |
            |  \          1 + x /    /     2\        x       /     2\    x *\1 + x /     \          1 + x / |   x      x*\1 + x /   /     2\  |    /     2\  |
            \                        \1 + x /                \1 + x /                                       \                       \1 + x /  /    \1 + x /  /
−5xatan⁡(x)(−8x2log⁡(x)(x2+1)3−(log⁡(x)x2+1+atan⁡(x)x)3+3(log⁡(x)x2+1+atan⁡(x)x)(2xlog⁡(x)(x2+1)2−2x(x2+1)+atan⁡(x)x2)+2log⁡(x)(x2+1)2+6(x2+1)2+3x2(x2+1)−2atan⁡(x)x3)- 5 x^{\operatorname{atan}{\left(x \right)}} \left(- \frac{8 x^{2} \log{\left(x \right)}}{\left(x^{2} + 1\right)^{3}} - \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)^{3} + 3 \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right) \left(\frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{2}{x \left(x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}\right) + \frac{2 \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{6}{\left(x^{2} + 1\right)^{2}} + \frac{3}{x^{2} \left(x^{2} + 1\right)} - \frac{2 \operatorname{atan}{\left(x \right)}}{x^{3}}\right)
Simplify
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