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How to use it?
- Derivative of:
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Derivative of x^(sin(x)^(2))
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Derivative of x^3-6*x^2
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Derivative of x^3-5x^2+x
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Derivative of (x+3)^2(x+5)-1
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Identical expressions
- (sqrt(3x+ two))^arctg(3x)
- ( square root of (3x plus 2)) to the power of arctg(3x)
- ( square root of (3x plus two)) to the power of arctg(3x)
- (√(3x+2))^arctg(3x)
- (sqrt(3x+2))arctg(3x)
- sqrt3x+2arctg3x
- sqrt3x+2^arctg3x
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Similar expressions
- (sqrt(3x-2))^arctg(3x)
Derivative of (sqrt(3x+2))^arctg(3x)
The solution
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atan(3*x) _________ \/ 3*x + 2
$$\left(\sqrt{3 x + 2}\right)^{\operatorname{atan}{\left(3 x \right)}}$$
(sqrt(3*x + 2))^atan(3*x)
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(3*x)
--------- / / _________\ \
2 |3*log\\/ 3*x + 2 / 3*atan(3*x)|
(3*x + 2) *|------------------ + -----------|
| 2 2*(3*x + 2)|
\ 1 + 9*x /
$$\left(3 x + 2\right)^{\frac{\operatorname{atan}{\left(3 x \right)}}{2}} \left(\frac{3 \log{\left(\sqrt{3 x + 2} \right)}}{9 x^{2} + 1} + \frac{3 \operatorname{atan}{\left(3 x \right)}}{2 \left(3 x + 2\right)}\right)$$
The second derivative
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/ / / _________\\ \
| /log(2 + 3*x) atan(3*x)\ |atan(3*x) 2*log\\/ 2 + 3*x /| |
atan(3*x) | |------------ + ---------|*|--------- + ------------------| |
--------- | | 2 2 + 3*x | | 2 + 3*x 2 | / _________\|
2 | 1 atan(3*x) \ 1 + 9*x / \ 1 + 9*x / 6*x*log\\/ 2 + 3*x /|
9*(2 + 3*x) *|-------------------- - ------------ + ----------------------------------------------------------- - --------------------|
|/ 2\ 2 4 2 |
|\1 + 9*x /*(2 + 3*x) 2*(2 + 3*x) / 2\ |
\ \1 + 9*x / /
$$9 \left(3 x + 2\right)^{\frac{\operatorname{atan}{\left(3 x \right)}}{2}} \left(- \frac{6 x \log{\left(\sqrt{3 x + 2} \right)}}{\left(9 x^{2} + 1\right)^{2}} + \frac{\left(\frac{2 \log{\left(\sqrt{3 x + 2} \right)}}{9 x^{2} + 1} + \frac{\operatorname{atan}{\left(3 x \right)}}{3 x + 2}\right) \left(\frac{\log{\left(3 x + 2 \right)}}{9 x^{2} + 1} + \frac{\operatorname{atan}{\left(3 x \right)}}{3 x + 2}\right)}{4} + \frac{1}{\left(3 x + 2\right) \left(9 x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(3 x \right)}}{2 \left(3 x + 2\right)^{2}}\right)$$
The third derivative
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/ / / _________\\ / / _________\\ \
| /log(2 + 3*x) atan(3*x)\ |atan(3*x) 2 12*x*log\\/ 2 + 3*x /| |atan(3*x) 2*log\\/ 2 + 3*x /| /atan(3*x) 2 6*x*log(2 + 3*x)\ 2 / / _________\\ |
| |------------ + ---------|*|---------- - -------------------- + ---------------------| |--------- + ------------------|*|---------- - -------------------- + ----------------| /log(2 + 3*x) atan(3*x)\ |atan(3*x) 2*log\\/ 2 + 3*x /| |
atan(3*x) | | 2 2 + 3*x | | 2 / 2\ 2 | | 2 + 3*x 2 | | 2 / 2\ 2 | |------------ + ---------| *|--------- + ------------------| |
--------- | / _________\ \ 1 + 9*x / |(2 + 3*x) \1 + 9*x /*(2 + 3*x) / 2\ | \ 1 + 9*x / |(2 + 3*x) \1 + 9*x /*(2 + 3*x) / 2\ | | 2 2 + 3*x | | 2 + 3*x 2 | 2 / _________\|
2 |atan(3*x) 2*log\\/ 2 + 3*x / 3 \ \1 + 9*x / / \ \1 + 9*x / / \ 1 + 9*x / \ 1 + 9*x / 9*x 72*x *log\\/ 2 + 3*x /|
27*(2 + 3*x) *|---------- - ------------------ - ----------------------- - -------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------- + ------------------------------------------------------------ - --------------------- + ----------------------|
| 3 2 / 2\ 2 2 4 8 2 3 |
|(2 + 3*x) / 2\ 2*\1 + 9*x /*(2 + 3*x) / 2\ / 2\ |
\ \1 + 9*x / \1 + 9*x / *(2 + 3*x) \1 + 9*x / /
$$27 \left(3 x + 2\right)^{\frac{\operatorname{atan}{\left(3 x \right)}}{2}} \left(\frac{72 x^{2} \log{\left(\sqrt{3 x + 2} \right)}}{\left(9 x^{2} + 1\right)^{3}} - \frac{9 x}{\left(3 x + 2\right) \left(9 x^{2} + 1\right)^{2}} + \frac{\left(\frac{2 \log{\left(\sqrt{3 x + 2} \right)}}{9 x^{2} + 1} + \frac{\operatorname{atan}{\left(3 x \right)}}{3 x + 2}\right) \left(\frac{\log{\left(3 x + 2 \right)}}{9 x^{2} + 1} + \frac{\operatorname{atan}{\left(3 x \right)}}{3 x + 2}\right)^{2}}{8} - \frac{\left(\frac{2 \log{\left(\sqrt{3 x + 2} \right)}}{9 x^{2} + 1} + \frac{\operatorname{atan}{\left(3 x \right)}}{3 x + 2}\right) \left(\frac{6 x \log{\left(3 x + 2 \right)}}{\left(9 x^{2} + 1\right)^{2}} - \frac{2}{\left(3 x + 2\right) \left(9 x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(3 x \right)}}{\left(3 x + 2\right)^{2}}\right)}{4} - \frac{\left(\frac{\log{\left(3 x + 2 \right)}}{9 x^{2} + 1} + \frac{\operatorname{atan}{\left(3 x \right)}}{3 x + 2}\right) \left(\frac{12 x \log{\left(\sqrt{3 x + 2} \right)}}{\left(9 x^{2} + 1\right)^{2}} - \frac{2}{\left(3 x + 2\right) \left(9 x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(3 x \right)}}{\left(3 x + 2\right)^{2}}\right)}{2} - \frac{2 \log{\left(\sqrt{3 x + 2} \right)}}{\left(9 x^{2} + 1\right)^{2}} - \frac{3}{2 \left(3 x + 2\right)^{2} \left(9 x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(3 x \right)}}{\left(3 x + 2\right)^{3}}\right)$$