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The derivative of the function/ x^3sinxcosx

Derivative of x^3sinxcosx

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
 3              
x *sin(x)*cos(x)
x3sin⁡(x)cos⁡(x)x^{3} \sin{\left(x \right)} \cos{\left(x \right)} 
(x^3*sin(x))*cos(x)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=x3sin⁡(x)f{\left(x \right)} = x^{3} \sin{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=x3f{\left(x \right)} = x^{3}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      The result is: x3cos⁡(x)+3x2sin⁡(x)x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}

    g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    The result is: −x3sin⁡2(x)+(x3cos⁡(x)+3x2sin⁡(x))cos⁡(x)- x^{3} \sin^{2}{\left(x \right)} + \left(x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}\right) \cos{\left(x \right)}

  2. Now simplify:

    x2(xcos⁡(2x)+3sin⁡(2x)2)x^{2} \left(x \cos{\left(2 x \right)} + \frac{3 \sin{\left(2 x \right)}}{2}\right)

The answer is:

x2(xcos⁡(2x)+3sin⁡(2x)2)x^{2} \left(x \cos{\left(2 x \right)} + \frac{3 \sin{\left(2 x \right)}}{2}\right)

The graph
02468-8-6-4-2-1010-20002000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/ 3             2       \           3    2   
\x *cos(x) + 3*x *sin(x)/*cos(x) - x *sin (x)
−x3sin⁡2(x)+(x3cos⁡(x)+3x2sin⁡(x))cos⁡(x)- x^{3} \sin^{2}{\left(x \right)} + \left(x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}\right) \cos{\left(x \right)}
Simplify
The second derivative [src] 
  //            2                    \           2                                                 \
x*\\6*sin(x) - x *sin(x) + 6*x*cos(x)/*cos(x) - x *cos(x)*sin(x) - 2*x*(3*sin(x) + x*cos(x))*sin(x)/
x(−x2sin⁡(x)cos⁡(x)−2x(xcos⁡(x)+3sin⁡(x))sin⁡(x)+(−x2sin⁡(x)+6xcos⁡(x)+6sin⁡(x))cos⁡(x))x \left(- x^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 2 x \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \sin{\left(x \right)} + \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right) \cos{\left(x \right)}\right)
Simplify
The third derivative [src] 
 3    2      /             3                           2       \              /            2                    \             2                             
x *sin (x) - \-6*sin(x) + x *cos(x) - 18*x*cos(x) + 9*x *sin(x)/*cos(x) - 3*x*\6*sin(x) - x *sin(x) + 6*x*cos(x)/*sin(x) - 3*x *(3*sin(x) + x*cos(x))*cos(x)
x3sin⁡2(x)−3x2(xcos⁡(x)+3sin⁡(x))cos⁡(x)−3x(−x2sin⁡(x)+6xcos⁡(x)+6sin⁡(x))sin⁡(x)−(x3cos⁡(x)+9x2sin⁡(x)−18xcos⁡(x)−6sin⁡(x))cos⁡(x)x^{3} \sin^{2}{\left(x \right)} - 3 x^{2} \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \cos{\left(x \right)} - 3 x \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right) \sin{\left(x \right)} - \left(x^{3} \cos{\left(x \right)} + 9 x^{2} \sin{\left(x \right)} - 18 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}\right) \cos{\left(x \right)}
Simplify
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