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y=x*cosx*sinx+1/2*cos^2x

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  • Derivative of:
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  • Derivative of x^2*e^x Derivative of x^2*e^x

  • Identical expressions

  • y=x*cosx*sinx+ one / two *cos^2x
  • y equally x multiply by co sinus of e of x multiply by sinus of x plus 1 divide by 2 multiply by co sinus of e of squared x
  • y equally x multiply by co sinus of e of x multiply by sinus of x plus one divide by two multiply by co sinus of e of squared x
  • y=x*cosx*sinx+1/2*cos2x
  • y=x*cosx*sinx+1/2*cos²x
  • y=x*cosx*sinx+1/2*cos to the power of 2x
  • y=xcosxsinx+1/2cos^2x
  • y=xcosxsinx+1/2cos2x
  • y=x*cosx*sinx+1 divide by 2*cos^2x

  • Similar expressions

  • y=xcosx*sinx+1/2cos^2x
  • y=x*cosx*sinx-1/2*cos^2x
The derivative of the function/ y=x*cosx*sinx+1/2*cos^2x

Derivative of y=x*cosx*sinx+1/2*cos^2x

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
                     2   
                  cos (x)
x*cos(x)*sin(x) + -------
                     2
$$x \cos{\left(x \right)} \sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{2}$$ 
(x*cos(x))*sin(x) + cos(x)^2/2
Detail solution

  1. Differentiate term by term:

    1. Apply the product rule:

      ; to find :

      1. Apply the product rule:

        ; to find :

        1. Apply the power rule: goes to

        ; to find :

        1. The derivative of cosine is negative sine:

        The result is:

      ; to find :

      1. The derivative of sine is cosine:

      The result is:

    2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of cosine is negative sine:

        The result of the chain rule is:

      So, the result is:

    The result is:

  2. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     2                                                 
x*cos (x) + (-x*sin(x) + cos(x))*sin(x) - cos(x)*sin(x)
$$x \cos^{2}{\left(x \right)} + \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}$$
Simplify
The second derivative [src] 
   2                                                                                    
sin (x) - (-cos(x) + x*sin(x))*cos(x) - (2*sin(x) + x*cos(x))*sin(x) - 2*x*cos(x)*sin(x)
$$- 2 x \sin{\left(x \right)} \cos{\left(x \right)} - \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)} - \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)} + \sin^{2}{\left(x \right)}$$
Simplify
The third derivative [src] 
                                                                     2                                              2   
(-cos(x) + x*sin(x))*sin(x) + (-3*cos(x) + x*sin(x))*sin(x) - 2*x*cos (x) - 2*(2*sin(x) + x*cos(x))*cos(x) + 2*x*sin (x)
$$2 x \sin^{2}{\left(x \right)} - 2 x \cos^{2}{\left(x \right)} + \left(x \sin{\left(x \right)} - 3 \cos{\left(x \right)}\right) \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)} - 2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}$$
Simplify
The graph
Derivative of y=x*cosx*sinx+1/2*cos^2x
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