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How to use it?
- Derivative of:
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Derivative of 7x
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Derivative of 4/x^3
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Derivative of 4*sqrt(x)
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Derivative of -(3*pi*cos(pi*t/6))/2
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Identical expressions
- (atan(3x))^(acot(x))
- ( arc tangent of gent of (3x)) to the power of ( arcco tangent of gent of (x))
- (atan(3x))(acot(x))
- atan3xacotx
- atan3x^acotx
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Similar expressions
- (arctan(3x))^(arccot(x))
- (atan(3x))^(arccotx)
Derivative of (atan(3x))^(acot(x))
The solution
You have entered
[src]
acot(x) atan (3*x)
$$\operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$
atan(3*x)^acot(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
[src]
acot(x) / log(atan(3*x)) 3*acot(x) \
atan (3*x)*|- -------------- + --------------------|
| 2 / 2\ |
\ 1 + x \1 + 9*x /*atan(3*x)/
$$\left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right) \operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$
The second derivative
[src]
/ 2 \
acot(x) |/ log(atan(3*x)) 3*acot(x) \ 9*acot(x) 6 2*x*log(atan(3*x)) 54*x*acot(x) |
atan (3*x)*||- -------------- + --------------------| - ---------------------- - ----------------------------- + ------------------ - ---------------------|
|| 2 / 2\ | 2 / 2\ / 2\ 2 2 |
|\ 1 + x \1 + 9*x /*atan(3*x)/ / 2\ 2 \1 + x /*\1 + 9*x /*atan(3*x) / 2\ / 2\ |
\ \1 + 9*x / *atan (3*x) \1 + x / \1 + 9*x / *atan(3*x)/
$$\left(- \frac{54 x \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} + \frac{2 x \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right)^{2} - \frac{9 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(3 x \right)}} - \frac{6}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}}\right) \operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$
The third derivative
[src]
/ 3 2 2 \
acot(x) |/ log(atan(3*x)) 3*acot(x) \ / log(atan(3*x)) 3*acot(x) \ / 2*x*log(atan(3*x)) 6 9*acot(x) 54*x*acot(x) \ 2*log(atan(3*x)) 54*acot(x) 8*x *log(atan(3*x)) 27 54*acot(x) 18*x 162*x 486*x*acot(x) 1944*x *acot(x) |
atan (3*x)*||- -------------- + --------------------| - 3*|- -------------- + --------------------|*|- ------------------ + ----------------------------- + ---------------------- + ---------------------| + ---------------- - --------------------- - ------------------- + ------------------------------- + ---------------------- + ------------------------------ + ------------------------------ + ---------------------- + ---------------------|
|| 2 / 2\ | | 2 / 2\ | | 2 / 2\ / 2\ 2 2 | 2 2 3 2 3 2 2 3 3 |
|\ 1 + x \1 + 9*x /*atan(3*x)/ \ 1 + x \1 + 9*x /*atan(3*x)/ | / 2\ \1 + x /*\1 + 9*x /*atan(3*x) / 2\ 2 / 2\ | / 2\ / 2\ / 2\ / 2\ / 2\ 2 / 2\ 3 / 2\ / 2\ / 2\ / 2\ / 2\ 2 / 2\ |
\ \ \1 + x / \1 + 9*x / *atan (3*x) \1 + 9*x / *atan(3*x)/ \1 + x / \1 + 9*x / *atan(3*x) \1 + x / \1 + x /*\1 + 9*x / *atan (3*x) \1 + 9*x / *atan (3*x) \1 + x / *\1 + 9*x /*atan(3*x) \1 + x /*\1 + 9*x / *atan(3*x) \1 + 9*x / *atan (3*x) \1 + 9*x / *atan(3*x)/
$$\left(\frac{1944 x^{2} \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{3} \operatorname{atan}{\left(3 x \right)}} - \frac{8 x^{2} \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{3}} + \frac{486 x \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{3} \operatorname{atan}^{2}{\left(3 x \right)}} + \frac{162 x}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} + \frac{18 x}{\left(x^{2} + 1\right)^{2} \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} + \left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right)^{3} - 3 \left(\frac{3 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{x^{2} + 1}\right) \left(\frac{54 x \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} - \frac{2 x \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{9 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(3 x \right)}} + \frac{6}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}}\right) - \frac{54 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{2} \operatorname{atan}{\left(3 x \right)}} + \frac{54 \operatorname{acot}{\left(x \right)}}{\left(9 x^{2} + 1\right)^{3} \operatorname{atan}^{3}{\left(3 x \right)}} + \frac{27}{\left(x^{2} + 1\right) \left(9 x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(3 x \right)}} + \frac{2 \log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{\left(x^{2} + 1\right)^{2}}\right) \operatorname{atan}^{\operatorname{acot}{\left(x \right)}}{\left(3 x \right)}$$