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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 1/x^8
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Derivative of (x^2)/36
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Derivative of x^2*e^(3-2*x)*log(2)^x
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Derivative of (x-1)/(x^2+1)
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Identical expressions
- (x*sinx)^ln(x*sinx)
- (x multiply by sinus of x) to the power of ln(x multiply by sinus of x)
- (x*sinx)ln(x*sinx)
- x*sinxlnx*sinx
- (xsinx)^ln(xsinx)
- (xsinx)ln(xsinx)
- xsinxlnxsinx
- xsinx^lnxsinx
Derivative of (x*sinx)^ln(x*sinx)
The solution
You have entered
[src]
log(x*sin(x)) (x*sin(x))
$$\left(x \sin{\left(x \right)}\right)^{\log{\left(x \sin{\left(x \right)} \right)}}$$
(x*sin(x))^log(x*sin(x))
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
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log(x*sin(x))
2*(x*sin(x)) *(x*cos(x) + sin(x))*log(x*sin(x))
-----------------------------------------------------------
x*sin(x)
$$\frac{2 \left(x \sin{\left(x \right)}\right)^{\log{\left(x \sin{\left(x \right)} \right)}} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}$$
The second derivative
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/ 2 2 2 \
log(x*sin(x)) | (x*cos(x) + sin(x)) (x*cos(x) + sin(x))*log(x*sin(x)) (x*cos(x) + sin(x))*cos(x)*log(x*sin(x)) 2*(x*cos(x) + sin(x)) *log (x*sin(x))|
2*(x*sin(x)) *|-(-2*cos(x) + x*sin(x))*log(x*sin(x)) + -------------------- - --------------------------------- - ---------------------------------------- + -------------------------------------|
\ x*sin(x) x sin(x) x*sin(x) /
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x*sin(x)
$$\frac{2 \left(x \sin{\left(x \right)}\right)^{\log{\left(x \sin{\left(x \right)} \right)}} \left(- \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \log{\left(x \sin{\left(x \right)} \right)}^{2}}{x \sin{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2}}{x \sin{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)}}{x}\right)}{x \sin{\left(x \right)}}$$
The third derivative
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/ 2 2 2 2 2 3 3 3 2 2 2 \
log(x*sin(x)) | 3*(x*cos(x) + sin(x)) 2*(-2*cos(x) + x*sin(x))*log(x*sin(x)) 2*(x*cos(x) + sin(x))*log(x*sin(x)) 6*(x*cos(x) + sin(x)) *log (x*sin(x)) 3*(x*cos(x) + sin(x)) *cos(x) 3*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x)) 2*cos (x)*(x*cos(x) + sin(x))*log(x*sin(x)) 2*(-2*cos(x) + x*sin(x))*cos(x)*log(x*sin(x)) 4*(x*cos(x) + sin(x)) *log (x*sin(x)) 6*(x*cos(x) + sin(x)) *log(x*sin(x)) 6*(x*cos(x) + sin(x)) *log (x*sin(x))*cos(x) 6*log (x*sin(x))*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x)) 2*(x*cos(x) + sin(x))*cos(x)*log(x*sin(x))|
2*(x*sin(x)) *|(x*cos(x) + sin(x))*log(x*sin(x)) - (3*sin(x) + x*cos(x))*log(x*sin(x)) - ---------------------- + -------------------------------------- + ----------------------------------- - ------------------------------------- - ----------------------------- - -------------------------------------------- + ------------------------------------------- + --------------------------------------------- + ------------------------------------- + ------------------------------------ - -------------------------------------------- - ----------------------------------------------------------- + ------------------------------------------|
| 2 x 2 2 2 x*sin(x) 2 sin(x) 2 2 2 2 2 x*sin(x) x*sin(x) |
\ x *sin(x) x x *sin(x) x*sin (x) sin (x) x *sin (x) x *sin (x) x*sin (x) /
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x*sin(x)
$$\frac{2 \left(x \sin{\left(x \right)}\right)^{\log{\left(x \sin{\left(x \right)} \right)}} \left(\frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} - \frac{6 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)}^{2}}{x \sin{\left(x \right)}} - \frac{3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)}}{x} - \frac{6 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \log{\left(x \sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin{\left(x \right)}} + \frac{4 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \log{\left(x \sin{\left(x \right)} \right)}^{3}}{x^{2} \sin^{2}{\left(x \right)}} + \frac{6 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \log{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin^{2}{\left(x \right)}} - \frac{6 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \log{\left(x \sin{\left(x \right)} \right)}^{2}}{x^{2} \sin{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2}}{x^{2} \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)}}{x^{2}}\right)}{x \sin{\left(x \right)}}$$