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Identical expressions
(cos(2x- five))^arctg(5x)
( co sinus of e of (2x minus 5)) to the power of arctg(5x)
( co sinus of e of (2x minus five)) to the power of arctg(5x)
(cos(2x-5))arctg(5x)
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cos2x-5^arctg5x
Similar expressions
(cos(2x+5))^arctg(5x)

The derivative of the function/ (cos(2x-5))^arctg(5x)

Derivative of (cos(2x-5))^arctg(5x)

The solution

You have entered [src]

   atan(5*x)         
cos         (2*x - 5)

\cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}

d /   atan(5*x)         \
--\cos         (2*x - 5)/
dx

\frac{d}{d x} \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}

Transform

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   atan(5*x)          /5*log(cos(2*x - 5))   2*atan(5*x)*sin(2*x - 5)\
cos         (2*x - 5)*|------------------- - ------------------------|
                      |             2              cos(2*x - 5)      |
                      \     1 + 25*x                                 /

\left(- \frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} + \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}

Simplify

The second derivative [src]

                       /                                                    2                                                                             2                    \
   atan(5*x)           |/  5*log(cos(-5 + 2*x))   2*atan(5*x)*sin(-5 + 2*x)\                  250*x*log(cos(-5 + 2*x))        20*sin(-5 + 2*x)       4*sin (-5 + 2*x)*atan(5*x)|
cos         (-5 + 2*x)*||- -------------------- + -------------------------|  - 4*atan(5*x) - ------------------------ - ------------------------- - --------------------------|
                       ||               2               cos(-5 + 2*x)      |                                   2         /        2\                          2                |
                       |\       1 + 25*x                                   /                        /        2\          \1 + 25*x /*cos(-5 + 2*x)         cos (-5 + 2*x)      |
                       \                                                                            \1 + 25*x /                                                                /

\left(\left(\frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} - \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right)^{2} - \frac{4 \sin^{2}{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos^{2}{\left(2 x - 5 \right)}} - 4 \operatorname{atan}{\left(5 x \right)} - \frac{250 x \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{2}} - \frac{20 \sin{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right) \cos{\left(2 x - 5 \right)}}\right) \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}

Simplify

The third derivative [src]

                       /                                                      3                                                                                               /                   2                                                                           \             2                                                     3                              2                                                \
   atan(5*x)           |  /  5*log(cos(-5 + 2*x))   2*atan(5*x)*sin(-5 + 2*x)\        60      250*log(cos(-5 + 2*x))     /  5*log(cos(-5 + 2*x))   2*atan(5*x)*sin(-5 + 2*x)\ |              2*sin (-5 + 2*x)*atan(5*x)        10*sin(-5 + 2*x)       125*x*log(cos(-5 + 2*x))|       60*sin (-5 + 2*x)        16*atan(5*x)*sin(-5 + 2*x)   16*sin (-5 + 2*x)*atan(5*x)   25000*x *log(cos(-5 + 2*x))      1500*x*sin(-5 + 2*x)   |
cos         (-5 + 2*x)*|- |- -------------------- + -------------------------|  - --------- - ---------------------- + 6*|- -------------------- + -------------------------|*|2*atan(5*x) + -------------------------- + ------------------------- + ------------------------| - -------------------------- - -------------------------- - --------------------------- + --------------------------- + --------------------------|
                       |  |               2               cos(-5 + 2*x)      |            2                   2          |               2               cos(-5 + 2*x)      | |                       2                   /        2\                                  2      |   /        2\    2                   cos(-5 + 2*x)                    3                                      3                     2              |
                       |  \       1 + 25*x                                   /    1 + 25*x         /        2\           \       1 + 25*x                                   / |                    cos (-5 + 2*x)         \1 + 25*x /*cos(-5 + 2*x)         /        2\       |   \1 + 25*x /*cos (-5 + 2*x)                                       cos (-5 + 2*x)                 /        2\           /        2\               |
                       \                                                                           \1 + 25*x /                                                                \                                                                             \1 + 25*x /       /                                                                                                   \1 + 25*x /           \1 + 25*x / *cos(-5 + 2*x)/

\left(- \left(\frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} - \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right)^{3} + 6 \cdot \left(\frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} - \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right) \left(\frac{2 \sin^{2}{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos^{2}{\left(2 x - 5 \right)}} + 2 \operatorname{atan}{\left(5 x \right)} + \frac{125 x \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{2}} + \frac{10 \sin{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right) \cos{\left(2 x - 5 \right)}}\right) - \frac{16 \sin^{3}{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos^{3}{\left(2 x - 5 \right)}} - \frac{16 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} + \frac{25000 x^{2} \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{3}} + \frac{1500 x \sin{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right)^{2} \cos{\left(2 x - 5 \right)}} - \frac{60 \sin^{2}{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right) \cos^{2}{\left(2 x - 5 \right)}} - \frac{60}{25 x^{2} + 1} - \frac{250 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{2}}\right) \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}

Simplify

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Derivative of (cos(2x-5))^arctg(5x)

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The solution