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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 5-7x
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Derivative of 3*x^2-1/x^3
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Derivative of 3^(1/x)
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Derivative of (2*x-1)^2
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Identical expressions
- ln(ch(x))/ln(cox(x))
- ln(ch(x)) divide by ln(cox(x))
- lnchx/lncoxx
Derivative of ln(ch(x))/ln(cox(x))
The solution
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[src]
log(cosh(x)) ------------- log(cos(x)*x)
$$\frac{\log{\left(\cosh{\left(x \right)} \right)}}{\log{\left(x \cos{\left(x \right)} \right)}}$$
log(cosh(x))/log(cos(x)*x)
The first derivative
[src]
sinh(x) (-x*sin(x) + cos(x))*log(cosh(x))
--------------------- - ---------------------------------
cosh(x)*log(cos(x)*x) 2
x*cos(x)*log (cos(x)*x)
$$\frac{\sinh{\left(x \right)}}{\log{\left(x \cos{\left(x \right)} \right)} \cosh{\left(x \right)}} - \frac{\left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(\cosh{\left(x \right)} \right)}}{x \log{\left(x \cos{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}$$
The second derivative
[src]
/ 2\
| -cos(x) + x*sin(x) (-cos(x) + x*sin(x))*sin(x) 2*(-cos(x) + x*sin(x)) |
2 |2*sin(x) + x*cos(x) - ------------------ + --------------------------- + -----------------------|*log(cosh(x))
sinh (x) \ x cos(x) x*cos(x)*log(x*cos(x))/ 2*(-cos(x) + x*sin(x))*sinh(x)
1 - -------- + --------------------------------------------------------------------------------------------------------------- + ------------------------------
2 x*cos(x)*log(x*cos(x)) x*cos(x)*cosh(x)*log(x*cos(x))
cosh (x)
---------------------------------------------------------------------------------------------------------------------------------------------------------------
log(x*cos(x))
$$\frac{- \frac{\sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} + 1 + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sinh{\left(x \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} \cosh{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} - \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right) \log{\left(\cosh{\left(x \right)} \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}}}{\log{\left(x \cos{\left(x \right)} \right)}}$$
The third derivative
[src]
/ 2 \ / 2 \ / 2 2 3 2 \
| sinh (x)| | sinh (x)| |-cos(x) + x*sin(x) 2*sin(x) + x*cos(x) (2*sin(x) + x*cos(x))*sin(x) sin (x)*(-cos(x) + x*sin(x)) (-cos(x) + x*sin(x))*sin(x) 3*(-cos(x) + x*sin(x)) 3*(-cos(x) + x*sin(x)) 3*(-cos(x) + x*sin(x)) *sin(x) 3*(-cos(x) + x*sin(x))*(2*sin(x) + x*cos(x)) | / 2\
2*|-1 + --------|*sinh(x) 3*|-1 + --------|*(-cos(x) + x*sin(x)) 2*|------------------ - ------------------- + ---------------------------- + ---------------------------- - --------------------------- - ----------------------- + ------------------------- + ------------------------------ + -------------------------------------------- + cos(x)|*log(cosh(x)) | -cos(x) + x*sin(x) (-cos(x) + x*sin(x))*sin(x) 2*(-cos(x) + x*sin(x)) |
| 2 | | 2 | | 2 x cos(x) 2 x*cos(x) 2 2 2 2 2 x*cos(x)*log(x*cos(x)) | 3*|2*sin(x) + x*cos(x) - ------------------ + --------------------------- + -----------------------|*sinh(x)
\ cosh (x)/ \ cosh (x)/ \ x cos (x) x *cos(x)*log(x*cos(x)) x *cos (x)*log (x*cos(x)) x*cos (x)*log(x*cos(x)) / \ x cos(x) x*cos(x)*log(x*cos(x))/
------------------------- - -------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------
cosh(x) x*cos(x)*log(x*cos(x)) x*cos(x)*log(x*cos(x)) x*cos(x)*cosh(x)*log(x*cos(x))
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
log(x*cos(x))
$$\frac{\frac{2 \left(\frac{\sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - 1\right) \sinh{\left(x \right)}}{\cosh{\left(x \right)}} - \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\frac{\sinh^{2}{\left(x \right)}}{\cosh^{2}{\left(x \right)}} - 1\right)}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} + \frac{3 \left(x \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} - \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right) \sinh{\left(x \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} \cosh{\left(x \right)}} + \frac{2 \left(\frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \cos{\left(x \right)} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos^{2}{\left(x \right)}} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} - \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{3}}{x^{2} \log{\left(x \cos{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)}} - \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x^{2} \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}} + \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x^{2}}\right) \log{\left(\cosh{\left(x \right)} \right)}}{x \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)}}}{\log{\left(x \cos{\left(x \right)} \right)}}$$