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1/3x^3*sin2x
The derivative of the function/ 1/3x^3*sin2x

Derivative of 1/3x^3*sin2x

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

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The solution

You have entered [src] 
 3         
x *sin(2*x)
-----------
     3
x3sin⁡(2x)3\frac{x^{3} \sin{\left(2 x \right)}}{3} 
  / 3         \
d |x *sin(2*x)|
--|-----------|
dx\     3     /
ddxx3sin⁡(2x)3\frac{d}{d x} \frac{x^{3} \sin{\left(2 x \right)}}{3}
Transform
Detail solution

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

    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⁡(2x)g{\left(x \right)} = \sin{\left(2 x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Let u=2xu = 2 x.

      2. The derivative of sine is cosine:

        ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

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

          1. Apply the power rule: xx goes to 11

          So, the result is: 22

        The result of the chain rule is:

        2cos⁡(2x)2 \cos{\left(2 x \right)}

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

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                 3         
 2            2*x *cos(2*x)
x *sin(2*x) + -------------
                    3
2x3cos⁡(2x)3+x2sin⁡(2x)\frac{2 x^{3} \cos{\left(2 x \right)}}{3} + x^{2} \sin{\left(2 x \right)}
Simplify
The second derivative [src] 
    /                  2                    \
    |               2*x *sin(2*x)           |
2*x*|2*x*cos(2*x) - ------------- + sin(2*x)|
    \                     3                 /
2x(−2x2sin⁡(2x)3+2xcos⁡(2x)+sin⁡(2x))2 x \left(- \frac{2 x^{2} \sin{\left(2 x \right)}}{3} + 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}\right)
Simplify
The third derivative [src] 
  /                                    3                    \
  |     2                           4*x *cos(2*x)           |
2*|- 6*x *sin(2*x) + 6*x*cos(2*x) - ------------- + sin(2*x)|
  \                                       3                 /
2(−4x3cos⁡(2x)3−6x2sin⁡(2x)+6xcos⁡(2x)+sin⁡(2x))2 \left(- \frac{4 x^{3} \cos{\left(2 x \right)}}{3} - 6 x^{2} \sin{\left(2 x \right)} + 6 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}\right)
Simplify
The graph
Derivative of 1/3x^3*sin2x
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