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five *sin6x-x^ two *sin2x
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5*sin6x+x^2*sin2x

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

Derivative of 5sin6x-x^2sin2x

The solution

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              2         
5*sin(6*x) - x *sin(2*x)

$$- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)}$$

d /              2         \
--\5*sin(6*x) - x *sin(2*x)/
dx

$$\frac{d}{d x} \left(- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)}\right)$$

Transform

Detail solution

Differentiate $- x^{2} \sin{\left(2 x \right)} + 5 \sin{\left(6 x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 6 x$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 6 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $6$
    The result of the chain rule is:
    
    $6 \cos{\left(6 x \right)}$
  So, the result is: $30 \cos{\left(6 x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    $g{\left(x \right)} = \sin{\left(2 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 2 x$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    The result is: $2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)}$
  So, the result is: $- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)}$
The result is: $- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)} + 30 \cos{\left(6 x \right)}$

The answer is:

$- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)} + 30 \cos{\left(6 x \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

                                2         
30*cos(6*x) - 2*x*sin(2*x) - 2*x *cos(2*x)

$$- 2 x^{2} \cos{\left(2 x \right)} - 2 x \sin{\left(2 x \right)} + 30 \cos{\left(6 x \right)}$$

Simplify

The second derivative [src]

  /                                            2         \
2*\-sin(2*x) - 90*sin(6*x) - 4*x*cos(2*x) + 2*x *sin(2*x)/

$$2 \cdot \left(2 x^{2} \sin{\left(2 x \right)} - 4 x \cos{\left(2 x \right)} - \sin{\left(2 x \right)} - 90 \sin{\left(6 x \right)}\right)$$

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)/

$$4 \cdot \left(2 x^{2} \cos{\left(2 x \right)} + 6 x \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)} - 270 \cos{\left(6 x \right)}\right)$$

Simplify

The graph

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Derivative of 5sin6x-x^2sin2x

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The solution

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Derivative of 5*sin6x-x^2*sin2x

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The solution

Derivative of 5sin6x-x^2sin2x