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How to use it?
- Derivative of:
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Derivative of (x-2)^2
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Derivative of e^-x
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Derivative of (x-1)^2
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Derivative of 1/sqrt(x)
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Identical expressions
- sqrt(two)*arctg(sqrt(two)*sinh(arccosh(x/sqrt(two))))+arctg(sinh(two *arccosh(x/sqrt(two))))
- square root of (2) multiply by arctg( square root of (2) multiply by hyperbolic sinus of e of (arc hyperbolic co sinus of e of ine of (x divide by square root of (2)))) plus arctg( hyperbolic sinus of e of (2 multiply by arc hyperbolic co sinus of e of ine of (x divide by square root of (2))))
- square root of (two) multiply by arctg( square root of (two) multiply by hyperbolic sinus of e of (arc hyperbolic co sinus of e of ine of (x divide by square root of (two)))) plus arctg( hyperbolic sinus of e of (two multiply by arc hyperbolic co sinus of e of ine of (x divide by square root of (two))))
- √(2)*arctg(√(2)*sinh(arccosh(x/√(2))))+arctg(sinh(2*arccosh(x/√(2))))
- sqrt(2)arctg(sqrt(2)sinh(arccosh(x/sqrt(2))))+arctg(sinh(2arccosh(x/sqrt(2))))
- sqrt2arctgsqrt2sinharccoshx/sqrt2+arctgsinh2arccoshx/sqrt2
- sqrt(2)*arctg(sqrt(2)*sinh(arccosh(x divide by sqrt(2))))+arctg(sinh(2*arccosh(x divide by sqrt(2))))
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Similar expressions
- sqrt(2)*arctg(sqrt(2)*sinh(arccosh(x/sqrt(2))))-arctg(sinh(2*arccosh(x/sqrt(2))))
The derivative of the function/
sqrt(2)*arctg(sqrt(2)*sinh(arccosh(x/sqrt(2))))+arctg(sinh(2*arccosh(x/sqrt(2))))
Derivative of sqrt(2)*arctg(sqrt(2)*sinh(arccosh(x/sqrt(2))))+arctg(sinh(2*arccosh(x/sqrt(2))))
The solution
You have entered
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___ / ___ / / x \\\ / / / x \\\
\/ 2 *atan|\/ 2 *sinh|acosh|-----||| + atan|sinh|2*acosh|-----|||
| | | ___||| | | | ___|||
\ \ \\/ 2 /// \ \ \\/ 2 ///
$$\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \sinh{\left(\operatorname{acosh}{\left(\frac{x}{\sqrt{2}} \right)} \right)} \right)} + \operatorname{atan}{\left(\sinh{\left(2 \operatorname{acosh}{\left(\frac{x}{\sqrt{2}} \right)} \right)} \right)}$$
sqrt(2)*atan(sqrt(2)*sinh(acosh(x/sqrt(2)))) + atan(sinh(2*acosh(x/sqrt(2))))
The first derivative
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___ / / x \\
\/ 2 *cosh|2*acosh|-----||
| | ___||
x \ \\/ 2 //
------------------------------------------ + ------------------------------------------
_________ _________
/ 2 / 2
/ 2/ / x \\\ / x / 2/ / x \\\ / x
|1 + 2*sinh |acosh|-----|||* / -1 + -- |1 + sinh |2*acosh|-----|||* / -1 + --
| | | ___||| \/ 2 | | | ___||| \/ 2
\ \ \\/ 2 /// \ \ \\/ 2 ///
$$\frac{x}{\sqrt{\frac{x^{2}}{2} - 1} \left(2 \sinh^{2}{\left(\operatorname{acosh}{\left(\frac{x}{\sqrt{2}} \right)} \right)} + 1\right)} + \frac{\sqrt{2} \cosh{\left(2 \operatorname{acosh}{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{\frac{x^{2}}{2} - 1} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{x}{\sqrt{2}} \right)} \right)} + 1\right)}$$
The second derivative
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_____________ ______________
/ / ___\\ / / ___\\ / / ___\\ / ___ / ___ / / ___\\
| |x*\/ 2 || 2| |x*\/ 2 || | |x*\/ 2 || 2 / x*\/ 2 / x*\/ 2 ___ | |x*\/ 2 ||
2 4*sinh|2*acosh|-------|| 8*cosh |2*acosh|-------||*sinh|2*acosh|-------|| 16*x * / 1 + ------- * / -1 + ------- x*\/ 2 *cosh|2*acosh|-------||
2 x \ \ 2 // \ \ 2 // \ \ 2 // \/ 2 \/ 2 \ \ 2 //
------------------------------------------------- - ----------------------------------------------- + --------------------------------------- - ------------------------------------------------ - --------------------------------------------- - --------------------------------------------
_________ 3/2 / / / ___\\\ 2 2 3/2
/ 2 / 2\ | 2| |x*\/ 2 ||| / 2\ / / / ___\\\ / 2\ / / ___\ / ___\\ / / / ___\\\ / 2\
/ x / / ___\ / ___\\ | x | / / ___\ / ___\\ |1 + sinh |2*acosh|-------|||*\-2 + x / | 2| |x*\/ 2 ||| / 2\ \-2 + x /*\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // | 2| |x*\/ 2 ||| | x |
/ -1 + -- *\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // |-1 + --| *\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // \ \ \ 2 /// |1 + sinh |2*acosh|-------||| *\-2 + x / 2*|1 + sinh |2*acosh|-------|||*|-1 + --|
\/ 2 \ 2 / \ \ \ 2 /// \ \ \ 2 /// \ 2 /
$$- \frac{16 x^{2} \sqrt{\frac{\sqrt{2} x}{2} - 1} \sqrt{\frac{\sqrt{2} x}{2} + 1}}{\left(x^{2} - 2\right) \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)^{2}} - \frac{x^{2}}{\left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)} - \frac{\sqrt{2} x \cosh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{2 \left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)} + \frac{4 \sinh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{\left(x^{2} - 2\right) \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)} - \frac{8 \sinh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} \cosh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{\left(x^{2} - 2\right) \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)^{2}} + \frac{2}{\sqrt{\frac{x^{2}}{2} - 1} \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)}$$
The third derivative
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_____________ ______________ _____________ ______________
/ / ___\\ / / ___\\ / / ___\\ / ___ / ___ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / ___ / ___ / / ___\\
| |x*\/ 2 || ___ 3| |x*\/ 2 || ___ | |x*\/ 2 || / x*\/ 2 / x*\/ 2 ___ 2| |x*\/ 2 || | |x*\/ 2 || ___ 3| |x*\/ 2 || 2| |x*\/ 2 || 2| |x*\/ 2 || | |x*\/ 2 || 3 / x*\/ 2 / x*\/ 2 ___ 2 | |x*\/ 2 ||
3 3 12*x*sinh|2*acosh|-------|| 4*\/ 2 *cosh |2*acosh|-------|| 3*\/ 2 *cosh|2*acosh|-------|| 48*x* / 1 + ------- * / -1 + ------- 12*\/ 2 *sinh |2*acosh|-------||*cosh|2*acosh|-------|| 16*\/ 2 *cosh |2*acosh|-------||*sinh |2*acosh|-------|| 3 / ___\ / ___\ 24*x*cosh |2*acosh|-------||*sinh|2*acosh|-------|| 48*x * / 1 + ------- * / -1 + ------- 3*\/ 2 *x *cosh|2*acosh|-------||
4*x 3*x 3*x \ \ 2 // \ \ 2 // \ \ 2 // \/ 2 \/ 2 \ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 // 16*x *\-2 + x*\/ 2 /*\2 + x*\/ 2 / \ \ 2 // \ \ 2 // \/ 2 \/ 2 \ \ 2 //
- ------------------------------------------------ - ----------------------------------------------- + ------------------------------------------------- - ---------------------------------------- - ------------------------------------------- + -------------------------------------------- - --------------------------------------------- - ------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------ + --------------------------------------------------- + ---------------------------------------------- + --------------------------------------------
3/2 3/2 5/2 / / / ___\\\ 2 2 3/2 3/2 2 2 3/2 3 3/2 3/2 2 2 2 5/2
/ 2\ 2 / 2\ / 2\ | 2| |x*\/ 2 ||| / 2\ / / / ___\\\ / 2\ / / / ___\\\ / 2\ / 2\ / / ___\ / ___\\ / / / ___\\\ / 2\ / / / ___\\\ / 2\ / 2\ 3 / / / ___\\\ 2 / 2\ / / ___\ / ___\\ / / / ___\\\ / 2\
| x | / / ___\ / ___\\ | x | / / ___\ / ___\\ | x | / / ___\ / ___\\ |1 + sinh |2*acosh|-------|||*\-2 + x / | 2| |x*\/ 2 ||| | x | | 2| |x*\/ 2 ||| | x | \-2 + x /*\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // | 2| |x*\/ 2 ||| | x | | 2| |x*\/ 2 ||| | x | | x | / / ___\ / ___\\ | 2| |x*\/ 2 ||| / 2\ \-2 + x / *\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // | 2| |x*\/ 2 ||| | x |
|-1 + --| *\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // |-1 + --| *\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // 2*|-1 + --| *\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // \ \ \ 2 /// |1 + sinh |2*acosh|-------||| *|-1 + --| 2*|1 + sinh |2*acosh|-------|||*|-1 + --| |1 + sinh |2*acosh|-------||| *|-1 + --| |1 + sinh |2*acosh|-------||| *|-1 + --| |-1 + --| *\2 + \-2 + x*\/ 2 /*\2 + x*\/ 2 // |1 + sinh |2*acosh|-------||| *\-2 + x / 4*|1 + sinh |2*acosh|-------|||*|-1 + --|
\ 2 / \ 2 / \ 2 / \ \ \ 2 /// \ 2 / \ \ \ 2 /// \ 2 / \ \ \ 2 /// \ 2 / \ \ \ 2 /// \ 2 / \ 2 / \ \ \ 2 /// \ \ \ 2 /// \ 2 /
$$\frac{48 x^{3} \sqrt{\frac{\sqrt{2} x}{2} - 1} \sqrt{\frac{\sqrt{2} x}{2} + 1}}{\left(x^{2} - 2\right)^{2} \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)^{2}} + \frac{16 x^{3} \left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right)}{\left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)^{3}} - \frac{4 x^{3}}{\left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)^{2}} + \frac{3 x^{3}}{2 \left(\frac{x^{2}}{2} - 1\right)^{\frac{5}{2}} \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)} + \frac{3 \sqrt{2} x^{2} \cosh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{4 \left(\frac{x^{2}}{2} - 1\right)^{\frac{5}{2}} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)} - \frac{48 x \sqrt{\frac{\sqrt{2} x}{2} - 1} \sqrt{\frac{\sqrt{2} x}{2} + 1}}{\left(x^{2} - 2\right) \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)^{2}} - \frac{12 x \sinh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{\left(x^{2} - 2\right)^{2} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)} + \frac{24 x \sinh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} \cosh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{\left(x^{2} - 2\right)^{2} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)^{2}} - \frac{3 x}{\left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\left(\sqrt{2} x - 2\right) \left(\sqrt{2} x + 2\right) + 2\right)} + \frac{3 \sqrt{2} \cosh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{2 \left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)} - \frac{12 \sqrt{2} \sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} \cosh{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{\left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)^{2}} - \frac{4 \sqrt{2} \cosh^{3}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{\left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)^{2}} + \frac{16 \sqrt{2} \sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} \cosh^{3}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)}}{\left(\frac{x^{2}}{2} - 1\right)^{\frac{3}{2}} \left(\sinh^{2}{\left(2 \operatorname{acosh}{\left(\frac{\sqrt{2} x}{2} \right)} \right)} + 1\right)^{3}}$$