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- Derivative of a implicit defined function
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- Partial derivative of the function
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How to use it?
- Derivative of:
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Derivative of x^12
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Derivative of x^2*e^(3-2*x)*log(2)^x
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Derivative of -(x^2+324)/x
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Derivative of x^3-1/5*x^2+2*x-4
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Identical expressions
- y=sqrt(x)^((sin^ two)x)
- y equally square root of (x) to the power of (( sinus of squared )x)
- y equally square root of (x) to the power of (( sinus of to the power of two)x)
- y=√(x)^((sin^2)x)
- y=sqrt(x)((sin2)x)
- y=sqrtxsin2x
- y=sqrt(x)^((sin²)x)
- y=sqrt(x) to the power of ((sin to the power of 2)x)
- y=sqrtx^sin^2x
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Similar expressions
- sqrt(x)^sin^2(x)
Derivative of y=sqrt(x)^((sin^2)x)
The solution
You have entered
[src]
2
sin (x)*x
___
\/ x
$$\left(\sqrt{x}\right)^{x \sin^{2}{\left(x \right)}}$$
(sqrt(x))^(sin(x)^2*x)
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
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Now simplify:
The answer is:
The first derivative
[src]
2
x*sin (x)
--------- / 2 \
2 |sin (x) / 2 \ / ___\|
x *|------- + \sin (x) + 2*x*cos(x)*sin(x)/*log\\/ x /|
\ 2 /
$$x^{\frac{x \sin^{2}{\left(x \right)}}{2}} \left(\left(2 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)}\right) \log{\left(\sqrt{x} \right)} + \frac{\sin^{2}{\left(x \right)}}{2}\right)$$
The second derivative
[src]
2
x*sin (x)
--------- / 2 / / ___\ \\
2 | / 2 2 \ / ___\ (2*x*cos(x) + sin(x))*sin(x) sin (x)*((2*x*cos(x) + sin(x))*log(x) + sin(x))*\2*(2*x*cos(x) + sin(x))*log\\/ x / + sin(x)/|
x *|cos(x)*sin(x) + 2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*log\\/ x / + ---------------------------- + ---------------------------------------------------------------------------------------------|
\ 2*x 4 /
$$x^{\frac{x \sin^{2}{\left(x \right)}}{2}} \left(\frac{\left(2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sqrt{x} \right)} + \sin{\left(x \right)}\right) \left(\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{4} + 2 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(\sqrt{x} \right)} + \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{2 x}\right)$$
The third derivative
[src]
2
x*sin (x) / / / 2 2 \ / ___\ (2*x*cos(x) + sin(x))*sin(x)\ / / ___\ \ / / 2 2 \ (2*x*cos(x) + sin(x))*sin(x)\ \
--------- | / 2 2 \ ((2*x*cos(x) + sin(x))*log(x) + sin(x))*|2*cos(x)*sin(x) + 4*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*log\\/ x / + ----------------------------|*sin(x) \2*(2*x*cos(x) + sin(x))*log\\/ x / + sin(x)/*|2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*log(x) + 2*cos(x)*sin(x) + ----------------------------|*sin(x) 2 3 / / ___\ \|
2 | 2 2 / 2 2 \ / ___\ 2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/ \ x / (2*x*cos(x) + sin(x))*sin(x) \ x / ((2*x*cos(x) + sin(x))*log(x) + sin(x)) *sin (x)*\2*(2*x*cos(x) + sin(x))*log\\/ x / + sin(x)/|
x *|cos (x) - sin (x) - 2*\- 3*cos (x) + 3*sin (x) + 4*x*cos(x)*sin(x)/*log\\/ x / + ------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------|
| x 2 2 4 8 |
\ 2*x /
$$x^{\frac{x \sin^{2}{\left(x \right)}}{2}} \left(\frac{\left(2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sqrt{x} \right)} + \sin{\left(x \right)}\right) \left(\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \sin^{3}{\left(x \right)}}{8} + \frac{\left(2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sqrt{x} \right)} + \sin{\left(x \right)}\right) \left(2 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{4} + \frac{\left(\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \sin{\left(x \right)}\right) \left(4 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(\sqrt{x} \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{2} - 2 \left(4 x \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \log{\left(\sqrt{x} \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{2 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{x} - \frac{\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{2 x^{2}}\right)$$
The graph