Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^5-5*x^3-20*x
Derivative of sin(x/5)
Derivative of sin(x)^(1/2)
Derivative of sin^2*2x
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
Similar expressions
sqrt(x)^sin^2(x)

The derivative of the function/ y=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

Don't know the steps in finding this derivative.

But the derivative is

$\left(x \sin^{2}{\left(x \right)}\right)^{x \sin^{2}{\left(x \right)}} \left(\log{\left(x \sin^{2}{\left(x \right)} \right)} + 1\right)$
Now simplify:

$\left(x \sin^{2}{\left(x \right)}\right)^{x \sin^{2}{\left(x \right)}} \left(\log{\left(x \sin^{2}{\left(x \right)} \right)} + 1\right)$

The answer is:

\left(x \sin^{2}{\left(x \right)}\right)^{x \sin^{2}{\left(x \right)}} \left(\log{\left(x \sin^{2}{\left(x \right)} \right)} + 1\right)

The graph

Plot the graph f(x)

Plot the graph f'(x)

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)

Simplify

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)

Simplify

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)

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=sqrt(x)^((sin^2)x)

The graph:

Piecewise:

The solution