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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 11*x-log(x+4)^11-3
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Derivative of y=x^3-8x^2+10x-4
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Derivative of y=12x+√x
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Derivative of x+ln(x^2-4)
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Identical expressions
- (xsinx)^sin(xsinx)
- (x sinus of x) to the power of sinus of (x sinus of x)
- (xsinx)sin(xsinx)
- xsinxsinxsinx
- xsinx^sinxsinx
Derivative of (xsinx)^sin(xsinx)
The solution
You have entered
[src]
sin(x*sin(x)) (x*sin(x))
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}}$$
(x*sin(x))^sin(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
[src]
sin(x*sin(x)) / (x*cos(x) + sin(x))*sin(x*sin(x))\
(x*sin(x)) *|(x*cos(x) + sin(x))*cos(x*sin(x))*log(x*sin(x)) + ---------------------------------|
\ x*sin(x) /
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}} \left(\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right)$$
The second derivative
[src]
/ 2 2 \
sin(x*sin(x)) | 2 / sin(x*sin(x))\ 2 (-2*cos(x) + x*sin(x))*sin(x*sin(x)) (x*cos(x) + sin(x))*sin(x*sin(x)) 2*(x*cos(x) + sin(x)) *cos(x*sin(x)) (x*cos(x) + sin(x))*cos(x)*sin(x*sin(x))|
(x*sin(x)) *|(x*cos(x) + sin(x)) *|cos(x*sin(x))*log(x*sin(x)) + -------------| - (x*cos(x) + sin(x)) *log(x*sin(x))*sin(x*sin(x)) - (-2*cos(x) + x*sin(x))*cos(x*sin(x))*log(x*sin(x)) - ------------------------------------ - --------------------------------- + ------------------------------------ - ----------------------------------------|
| \ x*sin(x) / x*sin(x) 2 x*sin(x) 2 |
\ x *sin(x) x*sin (x) /
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}} \left(- \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right)^{2} - \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \log{\left(x \sin{\left(x \right)} \right)} \sin{\left(x \sin{\left(x \right)} \right)} - \frac{\left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}}\right)$$
The third derivative
[src]
/ 3 / 2 \ 3 2 2 2 \
sin(x*sin(x)) | 3 / sin(x*sin(x))\ 3 / sin(x*sin(x))\ | 2 (-2*cos(x) + x*sin(x))*sin(x*sin(x)) (x*cos(x) + sin(x))*sin(x*sin(x)) 2*(x*cos(x) + sin(x)) *cos(x*sin(x)) (x*cos(x) + sin(x))*cos(x)*sin(x*sin(x))| (x*cos(x) + sin(x))*sin(x*sin(x)) (3*sin(x) + x*cos(x))*sin(x*sin(x)) 3*(x*cos(x) + sin(x)) *sin(x*sin(x)) 3*(x*cos(x) + sin(x)) *cos(x*sin(x)) 2*(x*cos(x) + sin(x))*sin(x*sin(x)) 2*(-2*cos(x) + x*sin(x))*sin(x*sin(x)) 6*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*cos(x*sin(x)) 3*(x*cos(x) + sin(x)) *cos(x)*cos(x*sin(x)) 2*cos (x)*(x*cos(x) + sin(x))*sin(x*sin(x)) 2*(-2*cos(x) + x*sin(x))*cos(x)*sin(x*sin(x)) 2*(x*cos(x) + sin(x))*cos(x)*sin(x*sin(x))|
(x*sin(x)) *|(x*cos(x) + sin(x)) *|cos(x*sin(x))*log(x*sin(x)) + -------------| - (x*cos(x) + sin(x)) *cos(x*sin(x))*log(x*sin(x)) - (3*sin(x) + x*cos(x))*cos(x*sin(x))*log(x*sin(x)) - 3*(x*cos(x) + sin(x))*|cos(x*sin(x))*log(x*sin(x)) + -------------|*|(x*cos(x) + sin(x)) *log(x*sin(x))*sin(x*sin(x)) + (-2*cos(x) + x*sin(x))*cos(x*sin(x))*log(x*sin(x)) + ------------------------------------ + --------------------------------- - ------------------------------------ + ----------------------------------------| + --------------------------------- - ----------------------------------- - ------------------------------------ - ------------------------------------ + ----------------------------------- + -------------------------------------- + 3*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*log(x*sin(x))*sin(x*sin(x)) - ---------------------------------------------------------- - ------------------------------------------- + ------------------------------------------- + --------------------------------------------- + ------------------------------------------|
| \ x*sin(x) / \ x*sin(x) / | x*sin(x) 2 x*sin(x) 2 | x*sin(x) x*sin(x) x*sin(x) 2 3 2 x*sin(x) 2 3 2 2 2 |
\ \ x *sin(x) x*sin (x) / x *sin(x) x *sin(x) x *sin(x) x*sin (x) x*sin (x) x*sin (x) x *sin (x) /
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}} \left(3 \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)} \sin{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right)^{3} - \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} - 3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{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)} \cos{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \log{\left(x \sin{\left(x \right)} \right)} \sin{\left(x \sin{\left(x \right)} \right)} + \frac{\left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}}\right) - \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\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) \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \right)} \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{x \sin^{3}{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x^{2} \sin^{2}{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{3} \sin{\left(x \right)}}\right)$$