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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
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Derivative of tan(x/2)
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Derivative of (x^2)
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Derivative of tan(x^3)
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Derivative of sin(x)/2
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Identical expressions
- arctg((two x- one)/(one +x^2))
- arctg((2x minus 1) divide by (1 plus x squared ))
- arctg((two x minus one) divide by (one plus x squared ))
- arctg((2x-1)/(1+x2))
- arctg2x-1/1+x2
- arctg((2x-1)/(1+x²))
- arctg((2x-1)/(1+x to the power of 2))
- arctg2x-1/1+x^2
- arctg((2x-1) divide by (1+x^2))
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Similar expressions
- arctg((2x-1)/(1-x^2))
- arctg((2x+1)/(1+x^2))
Derivative of arctg((2x-1)/(1+x^2))
The solution
You have entered
[src]
/2*x - 1\
atan|-------|
| 2|
\ 1 + x /
$$\operatorname{atan}{\left(\frac{2 x - 1}{x^{2} + 1} \right)}$$
atan((2*x - 1)/(1 + x^2))
The first derivative
[src]
2 2*x*(2*x - 1)
------ - -------------
2 2
1 + x / 2\
\1 + x /
----------------------
2
(2*x - 1)
1 + ----------
2
/ 2\
\1 + x /
$$\frac{- \frac{2 x \left(2 x - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x^{2} + 1}}{\frac{\left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} + 1}$$
The second derivative
[src]
/ 2 \
| / x*(-1 + 2*x)\ |
| 4*|-1 + ------------| *(-1 + 2*x)|
| 2 | 2 | |
| 4*x *(-1 + 2*x) \ 1 + x / |
-2*|-1 + 6*x - --------------- + ---------------------------------|
| 2 / 2\ |
| 1 + x / 2\ | (-1 + 2*x) | |
| \1 + x /*|1 + -----------| |
| | 2 | |
| | / 2\ | |
\ \ \1 + x / / /
-------------------------------------------------------------------
2 / 2\
/ 2\ | (-1 + 2*x) |
\1 + x / *|1 + -----------|
| 2 |
| / 2\ |
\ \1 + x / /
$$- \frac{2 \left(- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} + 1} + 6 x + \frac{4 \left(2 x - 1\right) \left(\frac{x \left(2 x - 1\right)}{x^{2} + 1} - 1\right)^{2}}{\left(x^{2} + 1\right) \left(\frac{\left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)} - 1\right)}{\left(x^{2} + 1\right)^{2} \left(\frac{\left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)}$$
The third derivative
[src]
/ / 2 2 2\ \
| 3 / x*(-1 + 2*x)\ | (-1 + 2*x) 8*x*(-1 + 2*x) 6*x *(-1 + 2*x) | / 2 \|
| 2 / x*(-1 + 2*x)\ 2*|-1 + ------------|*|2 - ----------- - -------------- + ----------------| / x*(-1 + 2*x)\ | 4*x *(-1 + 2*x)||
| 16*(-1 + 2*x) *|-1 + ------------| | 2 | | 2 2 2 | 4*(-1 + 2*x)*|-1 + ------------|*|-1 + 6*x - ---------------||
| 2 3 | 2 | \ 1 + x / | 1 + x 1 + x / 2\ | | 2 | | 2 ||
| 12*x 12*x *(-1 + 2*x) 6*x*(-1 + 2*x) \ 1 + x / \ \1 + x / / \ 1 + x / \ 1 + x /|
4*|-3 + ------ - ---------------- + -------------- - ----------------------------------- + --------------------------------------------------------------------------- - -------------------------------------------------------------|
| 2 2 2 2 / 2\ 2 / 2\ |
| 1 + x / 2\ 1 + x 3 / 2\ / 2\ | (-1 + 2*x) | / 2\ | (-1 + 2*x) | |
| \1 + x / / 2\ | (-1 + 2*x) | \1 + x /*|1 + -----------| \1 + x / *|1 + -----------| |
| \1 + x / *|1 + -----------| | 2 | | 2 | |
| | 2 | | / 2\ | | / 2\ | |
| | / 2\ | \ \1 + x / / \ \1 + x / / |
\ \ \1 + x / / /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 / 2\
/ 2\ | (-1 + 2*x) |
\1 + x / *|1 + -----------|
| 2 |
| / 2\ |
\ \1 + x / /
$$\frac{4 \left(- \frac{12 x^{3} \left(2 x - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{12 x^{2}}{x^{2} + 1} + \frac{6 x \left(2 x - 1\right)}{x^{2} + 1} - \frac{16 \left(2 x - 1\right)^{2} \left(\frac{x \left(2 x - 1\right)}{x^{2} + 1} - 1\right)^{3}}{\left(x^{2} + 1\right)^{3} \left(\frac{\left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)^{2}} - \frac{4 \left(2 x - 1\right) \left(\frac{x \left(2 x - 1\right)}{x^{2} + 1} - 1\right) \left(- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} + 1} + 6 x - 1\right)}{\left(x^{2} + 1\right)^{2} \left(\frac{\left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)} - 3 + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} + 1} - 1\right) \left(\frac{6 x^{2} \left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} - \frac{8 x \left(2 x - 1\right)}{x^{2} + 1} - \frac{\left(2 x - 1\right)^{2}}{x^{2} + 1} + 2\right)}{\left(x^{2} + 1\right) \left(\frac{\left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)}\right)}{\left(x^{2} + 1\right)^{2} \left(\frac{\left(2 x - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)}$$