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- Derivative of:
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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)
- cos2x-5arctg5x
- cos2x-5^arctg5x
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Similar expressions
- (cos(2x+5))^arctg(5x)
Derivative of (cos(2x-5))^arctg(5x)
The solution
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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
$$\frac{d}{d x} \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}$$
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
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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 /
$$\left(- \frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} + \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}$$
The second derivative
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/ 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 / /
$$\left(\left(\frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} - \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right)^{2} - \frac{4 \sin^{2}{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos^{2}{\left(2 x - 5 \right)}} - 4 \operatorname{atan}{\left(5 x \right)} - \frac{250 x \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{2}} - \frac{20 \sin{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right) \cos{\left(2 x - 5 \right)}}\right) \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}$$
The third derivative
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/ 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)/
$$\left(- \left(\frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} - \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right)^{3} + 6 \cdot \left(\frac{2 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} - \frac{5 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{25 x^{2} + 1}\right) \left(\frac{2 \sin^{2}{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos^{2}{\left(2 x - 5 \right)}} + 2 \operatorname{atan}{\left(5 x \right)} + \frac{125 x \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{2}} + \frac{10 \sin{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right) \cos{\left(2 x - 5 \right)}}\right) - \frac{16 \sin^{3}{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos^{3}{\left(2 x - 5 \right)}} - \frac{16 \sin{\left(2 x - 5 \right)} \operatorname{atan}{\left(5 x \right)}}{\cos{\left(2 x - 5 \right)}} + \frac{25000 x^{2} \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{3}} + \frac{1500 x \sin{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right)^{2} \cos{\left(2 x - 5 \right)}} - \frac{60 \sin^{2}{\left(2 x - 5 \right)}}{\left(25 x^{2} + 1\right) \cos^{2}{\left(2 x - 5 \right)}} - \frac{60}{25 x^{2} + 1} - \frac{250 \log{\left(\cos{\left(2 x - 5 \right)} \right)}}{\left(25 x^{2} + 1\right)^{2}}\right) \cos^{\operatorname{atan}{\left(5 x \right)}}{\left(2 x - 5 \right)}$$
The graph