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How to use it?
- Derivative of:
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Derivative of (x^2+1)^3
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Derivative of sin(cosx)
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Derivative of ln(-x)
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Derivative of sin(x/3)^(2)*cot(x/2)
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Identical expressions
- arcctg(x/ln(one +x^ two))
- arcctg(x divide by ln(1 plus x squared ))
- arcctg(x divide by ln(one plus x to the power of two))
- arcctg(x/ln(1+x2))
- arcctgx/ln1+x2
- arcctg(x/ln(1+x²))
- arcctg(x/ln(1+x to the power of 2))
- arcctgx/ln1+x^2
- arcctg(x divide by ln(1+x^2))
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Similar expressions
- arcctg(x/ln(1-x^2))
Derivative of arcctg(x/ln(1+x^2))
The solution
You have entered
[src]
/ x \
acot|-----------|
| / 2\|
\log\1 + x //
$$\operatorname{acot}{\left(\frac{x}{\log{\left(x^{2} + 1 \right)}} \right)}$$
acot(x/log(1 + x^2))
The first derivative
[src]
/ 2 \
| 1 2*x |
-|----------- - ---------------------|
| / 2\ / 2\ 2/ 2\|
\log\1 + x / \1 + x /*log \1 + x //
---------------------------------------
2
x
1 + ------------
2/ 2\
log \1 + x /
$$- \frac{- \frac{2 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}^{2}} + \frac{1}{\log{\left(x^{2} + 1 \right)}}}{\frac{x^{2}}{\log{\left(x^{2} + 1 \right)}^{2}} + 1}$$
The second derivative
[src]
/ 2 \
| 2 2 / 2 \ |
| 2*x 4*x | 2*x | |
| -3 + ------ + -------------------- |-1 + --------------------| |
| 2 / 2\ / 2\ | / 2\ / 2\| |
| 1 + x \1 + x /*log\1 + x / \ \1 + x /*log\1 + x // |
2*x*|- ---------------------------------- + ------------------------------|
| 2 / 2 \ |
| 1 + x | x | / 2\|
| |1 + ------------|*log\1 + x /|
| | 2/ 2\| |
\ \ log \1 + x // /
---------------------------------------------------------------------------
/ 2 \
| x | 2/ 2\
|1 + ------------|*log \1 + x /
| 2/ 2\|
\ log \1 + x //
$$\frac{2 x \left(\frac{\left(\frac{2 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 1\right)^{2}}{\left(\frac{x^{2}}{\log{\left(x^{2} + 1 \right)}^{2}} + 1\right) \log{\left(x^{2} + 1 \right)}} - \frac{\frac{2 x^{2}}{x^{2} + 1} + \frac{4 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 3}{x^{2} + 1}\right)}{\left(\frac{x^{2}}{\log{\left(x^{2} + 1 \right)}^{2}} + 1\right) \log{\left(x^{2} + 1 \right)}^{2}}$$
The third derivative
[src]
/ 2 4 2 4 4 / 2 \ / 2 4 4 \ 3 \
| 12*x 8*x 24*x 24*x 24*x | 2*x | | 10*x 4*x 12*x | / 2 \ / 2 \ / 2 2 \|
|3 - ------ + --------- - -------------------- + --------------------- + ---------------------- |-1 + --------------------|*|1 - -------------------- + --------------------- + ----------------------| 2 | 2*x | 2 | 2*x | | 2*x 4*x ||
| 2 2 / 2\ / 2\ 2 2 | / 2\ / 2\| | / 2\ / 2\ 2 2 | 4*x *|-1 + --------------------| 4*x *|-1 + --------------------|*|-3 + ------ + --------------------||
| 1 + x / 2\ \1 + x /*log\1 + x / / 2\ / 2\ / 2\ 2/ 2\ \ \1 + x /*log\1 + x // | \1 + x /*log\1 + x / / 2\ / 2\ / 2\ 2/ 2\| | / 2\ / 2\| | / 2\ / 2\| | 2 / 2\ / 2\||
| \1 + x / \1 + x / *log\1 + x / \1 + x / *log \1 + x / \ \1 + x / *log\1 + x / \1 + x / *log \1 + x // \ \1 + x /*log\1 + x // \ \1 + x /*log\1 + x // \ 1 + x \1 + x /*log\1 + x //|
2*|---------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------- + --------------------------------- - ---------------------------------------------------------------------|
| 2 / 2 \ 2 / 2 \ |
| 1 + x | x | / 2\ / 2 \ / 2\ | x | 2/ 2\ |
| |1 + ------------|*log\1 + x / | x | 3/ 2\ \1 + x /*|1 + ------------|*log \1 + x / |
| | 2/ 2\| |1 + ------------| *log \1 + x / | 2/ 2\| |
| \ log \1 + x // | 2/ 2\| \ log \1 + x // |
\ \ log \1 + x // /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2 \
| x | 2/ 2\
|1 + ------------|*log \1 + x /
| 2/ 2\|
\ log \1 + x //
$$\frac{2 \left(\frac{4 x^{2} \left(\frac{2 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 1\right)^{3}}{\left(\frac{x^{2}}{\log{\left(x^{2} + 1 \right)}^{2}} + 1\right)^{2} \log{\left(x^{2} + 1 \right)}^{3}} - \frac{4 x^{2} \left(\frac{2 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 1\right) \left(\frac{2 x^{2}}{x^{2} + 1} + \frac{4 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 3\right)}{\left(x^{2} + 1\right) \left(\frac{x^{2}}{\log{\left(x^{2} + 1 \right)}^{2}} + 1\right) \log{\left(x^{2} + 1 \right)}^{2}} - \frac{\left(\frac{2 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 1\right) \left(\frac{4 x^{4}}{\left(x^{2} + 1\right)^{2} \log{\left(x^{2} + 1 \right)}} + \frac{12 x^{4}}{\left(x^{2} + 1\right)^{2} \log{\left(x^{2} + 1 \right)}^{2}} - \frac{10 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} + 1\right)}{\left(\frac{x^{2}}{\log{\left(x^{2} + 1 \right)}^{2}} + 1\right) \log{\left(x^{2} + 1 \right)}} + \frac{\frac{8 x^{4}}{\left(x^{2} + 1\right)^{2}} + \frac{24 x^{4}}{\left(x^{2} + 1\right)^{2} \log{\left(x^{2} + 1 \right)}} + \frac{24 x^{4}}{\left(x^{2} + 1\right)^{2} \log{\left(x^{2} + 1 \right)}^{2}} - \frac{12 x^{2}}{x^{2} + 1} - \frac{24 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} + 3}{x^{2} + 1}\right)}{\left(\frac{x^{2}}{\log{\left(x^{2} + 1 \right)}^{2}} + 1\right) \log{\left(x^{2} + 1 \right)}^{2}}$$