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- Derivative of:
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Derivative of e^2*x
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Derivative of cos(x)^2
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Derivative of x^sin(x)
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Derivative of sin(x^3)
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Identical expressions
- arctg^19sqrt^x^ twenty
- arctg to the power of 19 square root of to the power of x squared 0
- arctg to the power of 19 square root of to the power of x to the power of twenty
- arctg^19√^x^20
- arctg19sqrtx20
- arctg^19sqrt^x²0
- arctg to the power of 19sqrt to the power of x to the power of 20
Derivative of arctg^19sqrt^x^20
The solution
You have entered
[src]
/ / 20\\
| \x /|
19 \2 /
atan (x)*t
$$t^{2^{x^{20}}} \operatorname{atan}^{19}{\left(x \right)}$$
atan(x)^19*t^(2^(x^20))
The first derivative
[src]
/ / 20\\
| \x /| / / 20\\
\2 / 18 / 20\ | \x /|
19*t *atan (x) \x / \2 / 19 19
---------------------- + 20*2 *t *x *atan (x)*log(2)*log(t)
2
1 + x
$$20 \cdot 2^{x^{20}} t^{2^{x^{20}}} x^{19} \log{\left(2 \right)} \log{\left(t \right)} \operatorname{atan}^{19}{\left(x \right)} + \frac{19 t^{2^{x^{20}}} \operatorname{atan}^{18}{\left(x \right)}}{x^{2} + 1}$$
The second derivative
[src]
/ / 20\\ / / 20\ \
| \x /| | / 20\ / / 20\ \ \x / 19 |
\2 / 17 | 19*(-9 + x*atan(x)) \x / 18 2 | 20 \x / 20 | 380*2 *x *atan(x)*log(2)*log(t)|
2*t *atan (x)*|- ------------------- + 10*2 *x *atan (x)*\19 + 20*x *log(2) + 20*2 *x *log(2)*log(t)/*log(2)*log(t) + ------------------------------------|
| 2 2 |
| / 2\ 1 + x |
\ \1 + x / /
$$2 t^{2^{x^{20}}} \left(\frac{380 \cdot 2^{x^{20}} x^{19} \log{\left(2 \right)} \log{\left(t \right)} \operatorname{atan}{\left(x \right)}}{x^{2} + 1} + 10 \cdot 2^{x^{20}} x^{18} \left(20 \cdot 2^{x^{20}} x^{20} \log{\left(2 \right)} \log{\left(t \right)} + 20 x^{20} \log{\left(2 \right)} + 19\right) \log{\left(2 \right)} \log{\left(t \right)} \operatorname{atan}^{2}{\left(x \right)} - \frac{19 \left(x \operatorname{atan}{\left(x \right)} - 9\right)}{\left(x^{2} + 1\right)^{2}}\right) \operatorname{atan}^{17}{\left(x \right)}$$
The third derivative
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/ / 2 2 \ \
| | 2 153 54*x*atan(x) 4*x *atan (x)| |
/ / 20\\ |19*|- atan (x) + ------ - ------------ + -------------| / 20\ / 20\ / / 20\ \ |
| \x /| | | 2 2 2 | / 20\ / 20 / 20\ / 20\ \ \x / 19 \x / 18 2 | 20 \x / 20 | |
\2 / 16 | \ 1 + x 1 + x 1 + x / \x / 17 3 | 40 2 20 2*x 40 2 2 \x / 20 \x / 40 2 | 1140*2 *x *(-9 + x*atan(x))*atan(x)*log(2)*log(t) 570*2 *x *atan (x)*\19 + 20*x *log(2) + 20*2 *x *log(2)*log(t)/*log(2)*log(t)|
2*t *atan (x)*|------------------------------------------------------- + 20*2 *x *atan (x)*\171 + 200*x *log (2) + 570*x *log(2) + 200*2 *x *log (2)*log (t) + 570*2 *x *log(2)*log(t) + 600*2 *x *log (2)*log(t)/*log(2)*log(t) - ------------------------------------------------------ + ----------------------------------------------------------------------------------------|
| 2 2 2 |
| / 2\ / 2\ 1 + x |
\ \1 + x / \1 + x / /
$$2 t^{2^{x^{20}}} \left(- \frac{1140 \cdot 2^{x^{20}} x^{19} \left(x \operatorname{atan}{\left(x \right)} - 9\right) \log{\left(2 \right)} \log{\left(t \right)} \operatorname{atan}{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{570 \cdot 2^{x^{20}} x^{18} \left(20 \cdot 2^{x^{20}} x^{20} \log{\left(2 \right)} \log{\left(t \right)} + 20 x^{20} \log{\left(2 \right)} + 19\right) \log{\left(2 \right)} \log{\left(t \right)} \operatorname{atan}^{2}{\left(x \right)}}{x^{2} + 1} + 20 \cdot 2^{x^{20}} x^{17} \left(200 \cdot 2^{2 x^{20}} x^{40} \log{\left(2 \right)}^{2} \log{\left(t \right)}^{2} + 600 \cdot 2^{x^{20}} x^{40} \log{\left(2 \right)}^{2} \log{\left(t \right)} + 570 \cdot 2^{x^{20}} x^{20} \log{\left(2 \right)} \log{\left(t \right)} + 200 x^{40} \log{\left(2 \right)}^{2} + 570 x^{20} \log{\left(2 \right)} + 171\right) \log{\left(2 \right)} \log{\left(t \right)} \operatorname{atan}^{3}{\left(x \right)} + \frac{19 \left(\frac{4 x^{2} \operatorname{atan}^{2}{\left(x \right)}}{x^{2} + 1} - \frac{54 x \operatorname{atan}{\left(x \right)}}{x^{2} + 1} - \operatorname{atan}^{2}{\left(x \right)} + \frac{153}{x^{2} + 1}\right)}{\left(x^{2} + 1\right)^{2}}\right) \operatorname{atan}^{16}{\left(x \right)}$$