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How to use it?
- Derivative of:
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Derivative of e^(2-x)
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Derivative of x^(4/5)
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Derivative of f(x)=5
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Derivative of 2/x²
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Identical expressions
- ln((x+ two - two *sqrt(x+ one))/(x+ two +sqrt(x+ one)))- two /sqrt(three)*arctg((one + two *sqrt(x+ one))/sqrt(three))
- ln((x plus 2 minus 2 multiply by square root of (x plus 1)) divide by (x plus 2 plus square root of (x plus 1))) minus 2 divide by square root of (3) multiply by arctg((1 plus 2 multiply by square root of (x plus 1)) divide by square root of (3))
- ln((x plus two minus two multiply by square root of (x plus one)) divide by (x plus two plus square root of (x plus one))) minus two divide by square root of (three) multiply by arctg((one plus two multiply by square root of (x plus one)) divide by square root of (three))
- ln((x+2-2*√(x+1))/(x+2+√(x+1)))-2/√(3)*arctg((1+2*√(x+1))/√(3))
- ln((x+2-2sqrt(x+1))/(x+2+sqrt(x+1)))-2/sqrt(3)arctg((1+2sqrt(x+1))/sqrt(3))
- lnx+2-2sqrtx+1/x+2+sqrtx+1-2/sqrt3arctg1+2sqrtx+1/sqrt3
- ln((x+2-2*sqrt(x+1)) divide by (x+2+sqrt(x+1)))-2 divide by sqrt(3)*arctg((1+2*sqrt(x+1)) divide by sqrt(3))
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Similar expressions
- ln((x+2-2*sqrt(x+1))/(x-2+sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
- ln((x+2-2*sqrt(x+1))/(x+2+sqrt(x-1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
- ln((x-2-2*sqrt(x+1))/(x+2+sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
- ln((x+2+2*sqrt(x+1))/(x+2+sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
- ln((x+2-2*sqrt(x+1))/(x+2-sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
- ln((x+2-2*sqrt(x+1))/(x+2+sqrt(x+1)))-2/sqrt(3)*arctg((1-2*sqrt(x+1))/sqrt(3))
- ln((x+2-2*sqrt(x-1))/(x+2+sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
- ln((x+2-2*sqrt(x+1))/(x+2+sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x-1))/sqrt(3))
- ln((x+2-2*sqrt(x+1))/(x+2+sqrt(x+1)))+2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
The derivative of the function/
ln((x+2-2*sqrt(x+1))/(x+2+sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
Derivative of ln((x+2-2*sqrt(x+1))/(x+2+sqrt(x+1)))-2/sqrt(3)*arctg((1+2*sqrt(x+1))/sqrt(3))
The solution
You have entered
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/ _______\ / _______\ |x + 2 - 2*\/ x + 1 | 2 |1 + 2*\/ x + 1 | log|-------------------| - -----*atan|---------------| | _______ | ___ | ___ | \ x + 2 + \/ x + 1 / \/ 3 \ \/ 3 /
$$\log{\left(\frac{- 2 \sqrt{x + 1} + \left(x + 2\right)}{\sqrt{x + 1} + \left(x + 2\right)} \right)} - \frac{2}{\sqrt{3}} \operatorname{atan}{\left(\frac{2 \sqrt{x + 1} + 1}{\sqrt{3}} \right)}$$
log((x + 2 - 2*sqrt(x + 1))/(x + 2 + sqrt(x + 1))) - 2/sqrt(3)*atan((1 + 2*sqrt(x + 1))/sqrt(3))
The first derivative
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/ 1 / 1 \ / _______\\
| 1 - --------- |-1 - -----------|*\x + 2 - 2*\/ x + 1 /|
| _______ | _______| |
| \/ x + 1 \ 2*\/ x + 1 / | / _______\
|----------------- + ----------------------------------------|*\x + 2 + \/ x + 1 /
| _______ 2 |
|x + 2 + \/ x + 1 / _______\ |
2 \ \x + 2 + \/ x + 1 / /
- ------------------------------------ + ----------------------------------------------------------------------------------
/ 2\ _______
| / _______\ | x + 2 - 2*\/ x + 1
| \1 + 2*\/ x + 1 / | _______
3*|1 + ------------------|*\/ x + 1
\ 3 /
$$\frac{\left(\frac{\left(-1 - \frac{1}{2 \sqrt{x + 1}}\right) \left(- 2 \sqrt{x + 1} + \left(x + 2\right)\right)}{\left(\sqrt{x + 1} + \left(x + 2\right)\right)^{2}} + \frac{1 - \frac{1}{\sqrt{x + 1}}}{\sqrt{x + 1} + \left(x + 2\right)}\right) \left(\sqrt{x + 1} + \left(x + 2\right)\right)}{- 2 \sqrt{x + 1} + \left(x + 2\right)} - \frac{2}{3 \sqrt{x + 1} \left(\frac{\left(2 \sqrt{x + 1} + 1\right)^{2}}{3} + 1\right)}$$
The second derivative
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2
/ 1 \ / _______\ / 1 \ / 1 \
2*|2 + ---------| *\2 + x - 2*\/ 1 + x / 4*|1 - ---------|*|2 + ---------| / / 1 \ / _______\\ / / 1 \ / _______\\
_______ | _______| | _______| | _______| | |2 + ---------|*\2 + x - 2*\/ 1 + x /| | |2 + ---------|*\2 + x - 2*\/ 1 + x /|
2 2 + x - 2*\/ 1 + x \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x / | | _______| | | | _______| |
- ---------- - ------------------------------ - ---------------------------------------- + --------------------------------- / 1 \ | 2 \ \/ 1 + x / | / 1 \ | 2 \ \/ 1 + x / |
3/2 3/2 / _______\ 2 _______ |1 - ---------|*|2 - --------- - -------------------------------------| |2 + ---------|*|2 - --------- - -------------------------------------|
(1 + x) (1 + x) *\2 + x + \/ 1 + x / / _______\ 2 + x + \/ 1 + x / _______\ | _______| | _______ _______ | | _______| | _______ _______ |
1 \2 + x + \/ 1 + x / 4*\1 + 2*\/ 1 + x / \ \/ 1 + x / \ \/ 1 + x 2 + x + \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x 2 + x + \/ 1 + x /
----------------------------------- - ---------------------------------------------------------------------------------------------------------------------------- + --------------------------------- - ----------------------------------------------------------------------- + -----------------------------------------------------------------------
/ 2\ / _______\ 2 2 / _______\ / _______\
3/2 | / _______\ | 4*\2 + x - 2*\/ 1 + x / / 2\ / _______\ 4*\2 + x + \/ 1 + x /*\2 + x - 2*\/ 1 + x /
(1 + x) *\3 + \1 + 2*\/ 1 + x / / | / _______\ | 2*\2 + x - 2*\/ 1 + x /
(1 + x)*\3 + \1 + 2*\/ 1 + x / /
$$- \frac{\left(1 - \frac{1}{\sqrt{x + 1}}\right) \left(- \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(x - 2 \sqrt{x + 1} + 2\right)}{x + \sqrt{x + 1} + 2} + 2 - \frac{2}{\sqrt{x + 1}}\right)}{2 \left(x - 2 \sqrt{x + 1} + 2\right)^{2}} + \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(- \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(x - 2 \sqrt{x + 1} + 2\right)}{x + \sqrt{x + 1} + 2} + 2 - \frac{2}{\sqrt{x + 1}}\right)}{4 \left(x - 2 \sqrt{x + 1} + 2\right) \left(x + \sqrt{x + 1} + 2\right)} - \frac{\frac{4 \left(1 - \frac{1}{\sqrt{x + 1}}\right) \left(2 + \frac{1}{\sqrt{x + 1}}\right)}{x + \sqrt{x + 1} + 2} - \frac{2 \left(2 + \frac{1}{\sqrt{x + 1}}\right)^{2} \left(x - 2 \sqrt{x + 1} + 2\right)}{\left(x + \sqrt{x + 1} + 2\right)^{2}} - \frac{x - 2 \sqrt{x + 1} + 2}{\left(x + 1\right)^{\frac{3}{2}} \left(x + \sqrt{x + 1} + 2\right)} - \frac{2}{\left(x + 1\right)^{\frac{3}{2}}}}{4 \left(x - 2 \sqrt{x + 1} + 2\right)} + \frac{4 \left(2 \sqrt{x + 1} + 1\right)}{\left(x + 1\right) \left(\left(2 \sqrt{x + 1} + 1\right)^{2} + 3\right)^{2}} + \frac{1}{\left(x + 1\right)^{\frac{3}{2}} \left(\left(2 \sqrt{x + 1} + 1\right)^{2} + 3\right)}$$
The third derivative
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/ 3 2 \ / 2 \ / 2 \
| / 1 \ / 1 \ / 1 \ / _______\ / 1 \ / 1 \ / 1 \ / _______\| | / 1 \ / _______\ / 1 \ / 1 \| | / 1 \ / _______\ / 1 \ / 1 \|
| 2*|2 + ---------| 2*|1 - ---------| 2*|2 + ---------| *\-2 - x + 2*\/ 1 + x / 4*|2 + ---------| *|1 - ---------| 2*|2 + ---------|*\-2 - x + 2*\/ 1 + x /| / / 1 \ / _______\\ | 2*|2 + ---------| *\2 + x - 2*\/ 1 + x / 4*|1 - ---------|*|2 + ---------|| / 1 \ / _______\ | 2*|2 + ---------| *\2 + x - 2*\/ 1 + x / 4*|1 - ---------|*|2 + ---------|| / 1 \ / _______\ / / 1 \ / _______\\
| _______ | _______| | _______| | _______| | _______| | _______| | _______| | | |2 + ---------|*\2 + x - 2*\/ 1 + x /| | _______ | _______| | _______| | _______|| |2 + ---------|*\2 + x - 2*\/ 1 + x / | _______ | _______| | _______| | _______|| |2 + ---------|*\2 + x - 2*\/ 1 + x / | |2 + ---------|*\2 + x - 2*\/ 1 + x /|
| 2 -2 - x + 2*\/ 1 + x \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x / | 2 | | _______| | / 1 \ | 2 2 + x - 2*\/ 1 + x \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x /| | _______| / 1 \ | 2 2 + x - 2*\/ 1 + x \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x /| | _______| | | _______| |
3*|- ---------- + ------------------------------ - ------------------------------ + ------------------------------ + ----------------------------------------- + ---------------------------------- + ----------------------------------------| / 1 \ | 2 \ \/ 1 + x / | |1 - ---------|*|- ---------- - ------------------------------ - ---------------------------------------- + ---------------------------------| 2 \ \/ 1 + x / |2 + ---------|*|- ---------- - ------------------------------ - ---------------------------------------- + ---------------------------------| 2 \ \/ 1 + x / / 1 \ / 1 \ | 2 \ \/ 1 + x / |
| 5/2 5/2 / _______\ 3/2 / _______\ 3/2 / _______\ 3 2 2 | |1 - ---------| *|2 - --------- - -------------------------------------| | _______| | 3/2 3/2 / _______\ 2 _______ | 2 2 - --------- - ------------------------------------- | _______| | 3/2 3/2 / _______\ 2 _______ | 2 - --------- - ------------------------------------- |1 - ---------|*|2 + ---------|*|2 - --------- - -------------------------------------|
| (1 + x) (1 + x) *\2 + x + \/ 1 + x / (1 + x) *\2 + x + \/ 1 + x / (1 + x) *\2 + x + \/ 1 + x / / _______\ / _______\ 3/2 / _______\ | | _______| | _______ _______ | \ \/ 1 + x / | (1 + x) (1 + x) *\2 + x + \/ 1 + x / / _______\ 2 + x + \/ 1 + x | / _______\ / _______\ _______ _______ \ \/ 1 + x / | (1 + x) (1 + x) *\2 + x + \/ 1 + x / / _______\ 2 + x + \/ 1 + x | _______ _______ | _______| | _______| | _______ _______ |
4 3 \ \2 + x + \/ 1 + x / \2 + x + \/ 1 + x / (1 + x) *\2 + x + \/ 1 + x / / \ \/ 1 + x / \ \/ 1 + x 2 + x + \/ 1 + x / \ \2 + x + \/ 1 + x / / 16*\1 + 2*\/ 1 + x / 6*\1 + 2*\/ 1 + x / \/ 1 + x 2 + x + \/ 1 + x \ \2 + x + \/ 1 + x / / \/ 1 + x 2 + x + \/ 1 + x \ \/ 1 + x / \ \/ 1 + x / \ \/ 1 + x 2 + x + \/ 1 + x /
------------------------------------ - ------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------ - ---------------------------------- - ----------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------ - ---------------------------------------------------------------------------------------
2 / 2\ / _______\ 3 2 3 2 2 / _______\ / _______\ 3/2 / _______\ / _______\ 2
/ 2\ 5/2 | / _______\ | 8*\2 + x - 2*\/ 1 + x / / _______\ / _______\ / 2\ / 2\ 3/2 / _______\ 4*\2 + x + \/ 1 + x /*\2 + x - 2*\/ 1 + x / 8*(1 + x) *\2 + x + \/ 1 + x /*\2 + x - 2*\/ 1 + x / / _______\ / _______\
3/2 | / _______\ | 2*(1 + x) *\3 + \1 + 2*\/ 1 + x / / \2 + x - 2*\/ 1 + x / 2*\2 + x - 2*\/ 1 + x / 3/2 | / _______\ | 2 | / _______\ | 4*(1 + x) *\2 + x - 2*\/ 1 + x / 2*\2 + x + \/ 1 + x /*\2 + x - 2*\/ 1 + x /
(1 + x) *\3 + \1 + 2*\/ 1 + x / / (1 + x) *\3 + \1 + 2*\/ 1 + x / / (1 + x) *\3 + \1 + 2*\/ 1 + x / /
$$\frac{\left(1 - \frac{1}{\sqrt{x + 1}}\right)^{2} \left(- \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(x - 2 \sqrt{x + 1} + 2\right)}{x + \sqrt{x + 1} + 2} + 2 - \frac{2}{\sqrt{x + 1}}\right)}{\left(x - 2 \sqrt{x + 1} + 2\right)^{3}} - \frac{\left(1 - \frac{1}{\sqrt{x + 1}}\right) \left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(- \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(x - 2 \sqrt{x + 1} + 2\right)}{x + \sqrt{x + 1} + 2} + 2 - \frac{2}{\sqrt{x + 1}}\right)}{2 \left(x - 2 \sqrt{x + 1} + 2\right)^{2} \left(x + \sqrt{x + 1} + 2\right)} + \frac{\left(1 - \frac{1}{\sqrt{x + 1}}\right) \left(\frac{4 \left(1 - \frac{1}{\sqrt{x + 1}}\right) \left(2 + \frac{1}{\sqrt{x + 1}}\right)}{x + \sqrt{x + 1} + 2} - \frac{2 \left(2 + \frac{1}{\sqrt{x + 1}}\right)^{2} \left(x - 2 \sqrt{x + 1} + 2\right)}{\left(x + \sqrt{x + 1} + 2\right)^{2}} - \frac{x - 2 \sqrt{x + 1} + 2}{\left(x + 1\right)^{\frac{3}{2}} \left(x + \sqrt{x + 1} + 2\right)} - \frac{2}{\left(x + 1\right)^{\frac{3}{2}}}\right)}{2 \left(x - 2 \sqrt{x + 1} + 2\right)^{2}} - \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(\frac{4 \left(1 - \frac{1}{\sqrt{x + 1}}\right) \left(2 + \frac{1}{\sqrt{x + 1}}\right)}{x + \sqrt{x + 1} + 2} - \frac{2 \left(2 + \frac{1}{\sqrt{x + 1}}\right)^{2} \left(x - 2 \sqrt{x + 1} + 2\right)}{\left(x + \sqrt{x + 1} + 2\right)^{2}} - \frac{x - 2 \sqrt{x + 1} + 2}{\left(x + 1\right)^{\frac{3}{2}} \left(x + \sqrt{x + 1} + 2\right)} - \frac{2}{\left(x + 1\right)^{\frac{3}{2}}}\right)}{4 \left(x - 2 \sqrt{x + 1} + 2\right) \left(x + \sqrt{x + 1} + 2\right)} + \frac{3 \left(\frac{4 \left(1 - \frac{1}{\sqrt{x + 1}}\right) \left(2 + \frac{1}{\sqrt{x + 1}}\right)^{2}}{\left(x + \sqrt{x + 1} + 2\right)^{2}} + \frac{2 \left(1 - \frac{1}{\sqrt{x + 1}}\right)}{\left(x + 1\right)^{\frac{3}{2}} \left(x + \sqrt{x + 1} + 2\right)} + \frac{2 \left(2 + \frac{1}{\sqrt{x + 1}}\right)^{3} \left(- x + 2 \sqrt{x + 1} - 2\right)}{\left(x + \sqrt{x + 1} + 2\right)^{3}} + \frac{2 \left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(- x + 2 \sqrt{x + 1} - 2\right)}{\left(x + 1\right)^{\frac{3}{2}} \left(x + \sqrt{x + 1} + 2\right)^{2}} - \frac{2 \left(2 + \frac{1}{\sqrt{x + 1}}\right)}{\left(x + 1\right)^{\frac{3}{2}} \left(x + \sqrt{x + 1} + 2\right)} + \frac{- x + 2 \sqrt{x + 1} - 2}{\left(x + 1\right)^{\frac{5}{2}} \left(x + \sqrt{x + 1} + 2\right)} - \frac{2}{\left(x + 1\right)^{\frac{5}{2}}}\right)}{8 \left(x - 2 \sqrt{x + 1} + 2\right)} - \frac{6 \left(2 \sqrt{x + 1} + 1\right)}{\left(x + 1\right)^{2} \left(\left(2 \sqrt{x + 1} + 1\right)^{2} + 3\right)^{2}} - \frac{16 \left(2 \sqrt{x + 1} + 1\right)^{2}}{\left(x + 1\right)^{\frac{3}{2}} \left(\left(2 \sqrt{x + 1} + 1\right)^{2} + 3\right)^{3}} - \frac{- \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(x - 2 \sqrt{x + 1} + 2\right)}{x + \sqrt{x + 1} + 2} + 2 - \frac{2}{\sqrt{x + 1}}}{8 \left(x + 1\right)^{\frac{3}{2}} \left(x - 2 \sqrt{x + 1} + 2\right) \left(x + \sqrt{x + 1} + 2\right)} - \frac{- \frac{\left(2 + \frac{1}{\sqrt{x + 1}}\right) \left(x - 2 \sqrt{x + 1} + 2\right)}{x + \sqrt{x + 1} + 2} + 2 - \frac{2}{\sqrt{x + 1}}}{4 \left(x + 1\right)^{\frac{3}{2}} \left(x - 2 \sqrt{x + 1} + 2\right)^{2}} + \frac{4}{\left(x + 1\right)^{\frac{3}{2}} \left(\left(2 \sqrt{x + 1} + 1\right)^{2} + 3\right)^{2}} - \frac{3}{2 \left(x + 1\right)^{\frac{5}{2}} \left(\left(2 \sqrt{x + 1} + 1\right)^{2} + 3\right)}$$