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How to use it?
- Derivative of:
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Derivative of y=4+6x-3x^4
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Derivative of √x+√x+√x
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Derivative of x*e^(-2x)
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Derivative of x+ln(x^2-4)
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Identical expressions
- ln(√(x^ two + one)/(chx))
- ln(√(x squared plus 1) divide by (chx))
- ln(√(x to the power of two plus one) divide by (chx))
- ln(√(x2+1)/(chx))
- ln√x2+1/chx
- ln(√(x²+1)/(chx))
- ln(√(x to the power of 2+1)/(chx))
- ln√x^2+1/chx
- ln(√(x^2+1) divide by (chx))
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Similar expressions
- ln(√(x^2-1)/(chx))
Derivative of ln(√(x^2+1)/(chx))
The solution
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/ ________\ | / 2 | |\/ x + 1 | log|-----------| \ cosh(x) /
$$\log{\left(\frac{\sqrt{x^{2} + 1}}{\cosh{\left(x \right)}} \right)}$$
log(sqrt(x^2 + 1)/cosh(x))
The first derivative
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/ ________ \
| / 2 |
| x \/ x + 1 *sinh(x)|
|------------------- - -------------------|*cosh(x)
| ________ 2 |
| / 2 cosh (x) |
\\/ x + 1 *cosh(x) /
---------------------------------------------------
________
/ 2
\/ x + 1
$$\frac{\left(\frac{x}{\sqrt{x^{2} + 1} \cosh{\left(x \right)}} - \frac{\sqrt{x^{2} + 1} \sinh{\left(x \right)}}{\cosh^{2}{\left(x \right)}}\right) \cosh{\left(x \right)}}{\sqrt{x^{2} + 1}}$$
The second derivative
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/ ________ \ / ________ \
| / 2 | | / 2 |
| x \/ 1 + x *sinh(x)| | x \/ 1 + x *sinh(x)|
|----------- - -------------------|*sinh(x) x*|----------- - -------------------|
| ________ cosh(x) | | ________ cosh(x) | ________
________ 2 | / 2 | | / 2 | / 2 2
1 / 2 x \\/ 1 + x / \\/ 1 + x / 2*\/ 1 + x *sinh (x) 2*x*sinh(x)
----------- - \/ 1 + x - ----------- + ------------------------------------------- - ------------------------------------- + ---------------------- - -------------------
________ 3/2 cosh(x) 2 2 ________
/ 2 / 2\ 1 + x cosh (x) / 2
\/ 1 + x \1 + x / \/ 1 + x *cosh(x)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
________
/ 2
\/ 1 + x
$$\frac{- \frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} - \frac{x \left(\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sqrt{x^{2} + 1} \sinh{\left(x \right)}}{\cosh{\left(x \right)}}\right)}{x^{2} + 1} - \frac{2 x \sinh{\left(x \right)}}{\sqrt{x^{2} + 1} \cosh{\left(x \right)}} + \frac{2 \sqrt{x^{2} + 1} \sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - \sqrt{x^{2} + 1} + \frac{\left(\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sqrt{x^{2} + 1} \sinh{\left(x \right)}}{\cosh{\left(x \right)}}\right) \sinh{\left(x \right)}}{\cosh{\left(x \right)}} + \frac{1}{\sqrt{x^{2} + 1}}}{\sqrt{x^{2} + 1}}$$
The third derivative
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________ / ________ \ / ________ \ / ________ \ / ________ \
/ 2 | ________ 2 / 2 2 | | ________ 2 / 2 2 | | / 2 | | / 2 |
x \/ 1 + x *sinh(x) | / 2 1 x 2*\/ 1 + x *sinh (x) 2*x*sinh(x) | | / 2 1 x 2*\/ 1 + x *sinh (x) 2*x*sinh(x) | 2 | x \/ 1 + x *sinh(x)| | x \/ 1 + x *sinh(x)|
----------- - ------------------- 2*|\/ 1 + x - ----------- + ----------- - ---------------------- + -------------------|*sinh(x) 2*x*|\/ 1 + x - ----------- + ----------- - ---------------------- + -------------------| 3*x *|----------- - -------------------| 2*x*|----------- - -------------------|*sinh(x)
________ cosh(x) ________ | ________ 3/2 2 ________ | | ________ 3/2 2 ________ | | ________ cosh(x) | ________ | ________ cosh(x) |
/ 2 3 / 2 3 | / 2 / 2\ cosh (x) / 2 | | / 2 / 2\ cosh (x) / 2 | | / 2 | / 2 2 2 | / 2 |
\/ 1 + x 3*x 2*x 3*x 6*\/ 1 + x *sinh (x) 3*sinh(x) \ \/ 1 + x \1 + x / \/ 1 + x *cosh(x)/ \ \/ 1 + x \1 + x / \/ 1 + x *cosh(x)/ \\/ 1 + x / 4*\/ 1 + x *sinh(x) 3*x *sinh(x) 6*x*sinh (x) \\/ 1 + x /
- --------------------------------- - ----------- - ----------- + ----------- - ---------------------- - ------------------- - -------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------- + ---------------------------------------- + --------------------- + ------------------- + -------------------- - -----------------------------------------------
2 3/2 ________ 5/2 3 ________ cosh(x) 2 2 cosh(x) 3/2 ________ / 2\
1 + x / 2\ / 2 / 2\ cosh (x) / 2 1 + x / 2\ / 2\ / 2 2 \1 + x /*cosh(x)
\1 + x / \/ 1 + x \1 + x / \/ 1 + x *cosh(x) \1 + x / \1 + x / *cosh(x) \/ 1 + x *cosh (x)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
________
/ 2
\/ 1 + x
$$\frac{\frac{3 x^{3}}{\left(x^{2} + 1\right)^{\frac{5}{2}}} + \frac{3 x^{2} \left(\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sqrt{x^{2} + 1} \sinh{\left(x \right)}}{\cosh{\left(x \right)}}\right)}{\left(x^{2} + 1\right)^{2}} + \frac{3 x^{2} \sinh{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}} \cosh{\left(x \right)}} - \frac{2 x \left(\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sqrt{x^{2} + 1} \sinh{\left(x \right)}}{\cosh{\left(x \right)}}\right) \sinh{\left(x \right)}}{\left(x^{2} + 1\right) \cosh{\left(x \right)}} + \frac{2 x \left(\frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{2 x \sinh{\left(x \right)}}{\sqrt{x^{2} + 1} \cosh{\left(x \right)}} - \frac{2 \sqrt{x^{2} + 1} \sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} + \sqrt{x^{2} + 1} - \frac{1}{\sqrt{x^{2} + 1}}\right)}{x^{2} + 1} + \frac{6 x \sinh^{2}{\left(x \right)}}{\sqrt{x^{2} + 1} \cosh^{2}{\left(x \right)}} - \frac{2 x}{\sqrt{x^{2} + 1}} - \frac{3 x}{\left(x^{2} + 1\right)^{\frac{3}{2}}} - \frac{6 \sqrt{x^{2} + 1} \sinh^{3}{\left(x \right)}}{\cosh^{3}{\left(x \right)}} + \frac{4 \sqrt{x^{2} + 1} \sinh{\left(x \right)}}{\cosh{\left(x \right)}} - \frac{2 \left(\frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{2 x \sinh{\left(x \right)}}{\sqrt{x^{2} + 1} \cosh{\left(x \right)}} - \frac{2 \sqrt{x^{2} + 1} \sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} + \sqrt{x^{2} + 1} - \frac{1}{\sqrt{x^{2} + 1}}\right) \sinh{\left(x \right)}}{\cosh{\left(x \right)}} - \frac{\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sqrt{x^{2} + 1} \sinh{\left(x \right)}}{\cosh{\left(x \right)}}}{x^{2} + 1} - \frac{3 \sinh{\left(x \right)}}{\sqrt{x^{2} + 1} \cosh{\left(x \right)}}}{\sqrt{x^{2} + 1}}$$