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5*sin6x-x^2*sin2x
The derivative of the function/ 5*sin6x-x^2*sin2x

Derivative of 5*sin6x-x^2*sin2x

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              2         
5*sin(6*x) - x *sin(2*x)
−x2sin⁡(2x)+5sin⁡(6x)- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)} 
d /              2         \
--\5*sin(6*x) - x *sin(2*x)/
dx
ddx(−x2sin⁡(2x)+5sin⁡(6x))\frac{d}{d x} \left(- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)}\right)
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Detail solution

  1. Differentiate −x2sin⁡(2x)+5sin⁡(6x)- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)} term by term:

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

      1. Let u=6xu = 6 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 ddx6x\frac{d}{d x} 6 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: 66

        The result of the chain rule is:

        6cos⁡(6x)6 \cos{\left(6 x \right)}

      So, the result is: 30cos⁡(6x)30 \cos{\left(6 x \right)}

    2. 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)=x2f{\left(x \right)} = x^{2}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Apply the power rule: x2x^{2} goes to 2x2 x

        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: 2x2cos⁡(2x)+2xsin⁡(2x)2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)}

      So, the result is: −2x2cos⁡(2x)−2xsin⁡(2x)- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)}

    The result is: −2x2cos⁡(2x)−2xsin⁡(2x)+30cos⁡(6x)- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)} + 30 \cos{\left(6 x \right)}

The answer is:

−2x2cos⁡(2x)−2xsin⁡(2x)+30cos⁡(6x)- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)} + 30 \cos{\left(6 x \right)}

The graph
02468-8-6-4-2-1010-500500
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The first derivative [src] 
                                2         
30*cos(6*x) - 2*x*sin(2*x) - 2*x *cos(2*x)
−2x2cos⁡(2x)−2xsin⁡(2x)+30cos⁡(6x)- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)} + 30 \cos{\left(6 x \right)}
Simplify
The second derivative [src] 
  /                                            2         \
2*\-sin(2*x) - 90*sin(6*x) - 4*x*cos(2*x) + 2*x *sin(2*x)/
2⋅(2x2sin⁡(2x)−4xcos⁡(2x)−sin⁡(2x)−90sin⁡(6x))2 \cdot \left(2 x^{2} \sin{\left(2 x \right)} - 4 x \cos{\left(2 x \right)} - \sin{\left(2 x \right)} - 90 \sin{\left(6 x \right)}\right)
Simplify
The third derivative [src] 
  /                                2                        \
4*\-270*cos(6*x) - 3*cos(2*x) + 2*x *cos(2*x) + 6*x*sin(2*x)/
4⋅(2x2cos⁡(2x)+6xsin⁡(2x)−3cos⁡(2x)−270cos⁡(6x))4 \cdot \left(2 x^{2} \cos{\left(2 x \right)} + 6 x \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)} - 270 \cos{\left(6 x \right)}\right)
Simplify
The graph
Derivative of 5*sin6x-x^2*sin2x
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