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The derivative of the function/ (tg(3x)-sin(3x))/2x^2

Derivative of (tg(3x)-sin(3x))/2x^2

The solution

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tan(3*x) - sin(3*x)  2
-------------------*x 
         2

$$x^{2} \frac{- \sin{\left(3 x \right)} + \tan{\left(3 x \right)}}{2}$$

((tan(3*x) - sin(3*x))/2)*x^2

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{2} \left(- \sin{\left(3 x \right)} + \tan{\left(3 x \right)}\right)$ and $g{\left(x \right)} = 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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 $2 x$
  $g{\left(x \right)} = - \sin{\left(3 x \right)} + \tan{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $- \sin{\left(3 x \right)} + \tan{\left(3 x \right)}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Let $u = 3 x$ .
      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 $\frac{d}{d x} 3 x$ :
        
        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: $3$
        
        The result of the chain rule is:
        
        $3 \cos{\left(3 x \right)}$
      So, the result is: $- 3 \cos{\left(3 x \right)}$
    2. Rewrite the function to be differentiated:
      
      $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
    3. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos{\left(3 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Let $u = 3 x$ .
      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 $\frac{d}{d x} 3 x$ :
        
        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: $3$
        
        The result of the chain rule is:
        
        $3 \cos{\left(3 x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Let $u = 3 x$ .
      2. The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
        
        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: $3$
        
        The result of the chain rule is:
        
        $- 3 \sin{\left(3 x \right)}$
      Now plug in to the quotient rule:
      
      $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
    The result is: $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - 3 \cos{\left(3 x \right)}$
  The result is: $x^{2} \left(\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - 3 \cos{\left(3 x \right)}\right) + 2 x \left(- \sin{\left(3 x \right)} + \tan{\left(3 x \right)}\right)$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $2$ is zero.
Now plug in to the quotient rule:

$\frac{x^{2} \left(\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - 3 \cos{\left(3 x \right)}\right)}{2} + x \left(- \sin{\left(3 x \right)} + \tan{\left(3 x \right)}\right)$
Now simplify:

$\frac{x \left(3 x \left(1 - \cos^{3}{\left(3 x \right)}\right) - 2 \left(\cos{\left(3 x \right)} - 1\right) \sin{\left(3 x \right)} \cos{\left(3 x \right)}\right)}{2 \cos^{2}{\left(3 x \right)}}$

The answer is:

$\frac{x \left(3 x \left(1 - \cos^{3}{\left(3 x \right)}\right) - 2 \left(\cos{\left(3 x \right)} - 1\right) \sin{\left(3 x \right)} \cos{\left(3 x \right)}\right)}{2 \cos^{2}{\left(3 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                             /                      2     \
                           2 |3   3*cos(3*x)   3*tan (3*x)|
x*(tan(3*x) - sin(3*x)) + x *|- - ---------- + -----------|
                             \2       2             2     /

$$x^{2} \left(- \frac{3 \cos{\left(3 x \right)}}{2} + \frac{3 \tan^{2}{\left(3 x \right)}}{2} + \frac{3}{2}\right) + x \left(- \sin{\left(3 x \right)} + \tan{\left(3 x \right)}\right)$$

Simplify

The second derivative [src]

                                                2 /  /       2     \                    \           
                /       2                \   9*x *\2*\1 + tan (3*x)/*tan(3*x) + sin(3*x)/           
-sin(3*x) + 6*x*\1 + tan (3*x) - cos(3*x)/ + -------------------------------------------- + tan(3*x)
                                                                  2

$$\frac{9 x^{2} \left(2 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} + \sin{\left(3 x \right)}\right)}{2} + 6 x \left(- \cos{\left(3 x \right)} + \tan^{2}{\left(3 x \right)} + 1\right) - \sin{\left(3 x \right)} + \tan{\left(3 x \right)}$$

Simplify

The third derivative [src]

  /                                                                              /                 2                                         \\
  |                                                                            2 |  /       2     \         2      /       2     \           ||
  |       2                       /  /       2     \                    \   3*x *\2*\1 + tan (3*x)/  + 4*tan (3*x)*\1 + tan (3*x)/ + cos(3*x)/|
9*|1 + tan (3*x) - cos(3*x) + 3*x*\2*\1 + tan (3*x)/*tan(3*x) + sin(3*x)/ + ------------------------------------------------------------------|
  \                                                                                                         2                                 /

$$9 \left(\frac{3 x^{2} \left(2 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{2} + 3 x \left(2 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} + \sin{\left(3 x \right)}\right) - \cos{\left(3 x \right)} + \tan^{2}{\left(3 x \right)} + 1\right)$$

Simplify

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Derivative of (tg(3x)-sin(3x))/2x^2

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