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Similar expressions
tan((9x^2+1)/3x)

The derivative of the function/ tan((9x^2-1)/3x)

Derivative of tan((9x^2-1)/3x)

The solution

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   /   2      \
   |9*x  - 1  |
tan|--------*x|
   \   3      /

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

tan(((9*x^2 - 1)/3)*x)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(x \frac{9 x^{2} - 1}{3} \right)} = \frac{\sin{\left(x \frac{9 x^{2} - 1}{3} \right)}}{\cos{\left(x \frac{9 x^{2} - 1}{3} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(x \frac{9 x^{2} - 1}{3} \right)}$ and $g{\left(x \right)} = \cos{\left(x \frac{9 x^{2} - 1}{3} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x \frac{9 x^{2} - 1}{3}$ .
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 $\frac{d}{d x} x \frac{9 x^{2} - 1}{3}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = x \left(9 x^{2} - 1\right)$ and $g{\left(x \right)} = 3$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = 9 x^{2} - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Differentiate $9 x^{2} - 1$ term by term:
        
        The derivative of the constant $-1$ is zero.
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $18 x$
        
        The result is: $18 x$
      The result is: $27 x^{2} - 1$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of the constant $3$ is zero.
    Now plug in to the quotient rule:
    
    $9 x^{2} - \frac{1}{3}$
  The result of the chain rule is:
  
  $\left(9 x^{2} - \frac{1}{3}\right) \cos{\left(x \frac{9 x^{2} - 1}{3} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x \frac{9 x^{2} - 1}{3}$ .
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 $\frac{d}{d x} x \frac{9 x^{2} - 1}{3}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = x \left(9 x^{2} - 1\right)$ and $g{\left(x \right)} = 3$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = 9 x^{2} - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Differentiate $9 x^{2} - 1$ term by term:
        
        The derivative of the constant $-1$ is zero.
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $18 x$
        
        The result is: $18 x$
      The result is: $27 x^{2} - 1$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of the constant $3$ is zero.
    Now plug in to the quotient rule:
    
    $9 x^{2} - \frac{1}{3}$
  The result of the chain rule is:
  
  $- \left(9 x^{2} - \frac{1}{3}\right) \sin{\left(x \frac{9 x^{2} - 1}{3} \right)}$
Now plug in to the quotient rule:

$\frac{\left(9 x^{2} - \frac{1}{3}\right) \sin^{2}{\left(x \frac{9 x^{2} - 1}{3} \right)} + \left(9 x^{2} - \frac{1}{3}\right) \cos^{2}{\left(x \frac{9 x^{2} - 1}{3} \right)}}{\cos^{2}{\left(x \frac{9 x^{2} - 1}{3} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/        /   2      \\ /          2    \
|       2|9*x  - 1  || |   2   9*x  - 1|
|1 + tan |--------*x||*|6*x  + --------|
\        \   3      // \          3    /

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

Simplify

The second derivative [src]

  /        /  /        2\\\                                          
  |       2|x*\-1 + 9*x /||                                          
  |    tan |-------------|| /                   2    /  /        2\\\
  |1       \      3      /| |       /         2\     |x*\-1 + 9*x /||
2*|- + -------------------|*|81*x + \-1 + 27*x / *tan|-------------||
  \9            9         / \                        \      3      //

$$2 \left(81 x + \left(27 x^{2} - 1\right)^{2} \tan{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)}\right) \left(\frac{\tan^{2}{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)}}{9} + \frac{1}{9}\right)$$

Simplify

The third derivative [src]

  /                                                     2                                                                                                                                               \
  |                            /        /  /        2\\\              3                 3     /  /        2\\ /        /  /        2\\\                                                                 |
  |                            |       2|x*\-1 + 9*x /||  /         2\      /         2\     2|x*\-1 + 9*x /| |       2|x*\-1 + 9*x /||                                                                 |
  |          /  /        2\\   |1 + tan |-------------|| *\-1 + 27*x /    2*\-1 + 27*x / *tan |-------------|*|1 + tan |-------------||        /        /  /        2\\\                 /  /        2\\|
  |         2|x*\-1 + 9*x /|   \        \      3      //                                      \      3      / \        \      3      //        |       2|x*\-1 + 9*x /|| /         2\    |x*\-1 + 9*x /||
2*|9 + 9*tan |-------------| + ---------------------------------------- + ------------------------------------------------------------- + 18*x*|1 + tan |-------------||*\-1 + 27*x /*tan|-------------||
  \          \      3      /                      27                                                    27                                     \        \      3      //                 \      3      //

$$2 \left(18 x \left(27 x^{2} - 1\right) \left(\tan^{2}{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)} + 1\right) \tan{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)} + \frac{\left(27 x^{2} - 1\right)^{3} \left(\tan^{2}{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)} + 1\right)^{2}}{27} + \frac{2 \left(27 x^{2} - 1\right)^{3} \left(\tan^{2}{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)} + 1\right) \tan^{2}{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)}}{27} + 9 \tan^{2}{\left(\frac{x \left(9 x^{2} - 1\right)}{3} \right)} + 9\right)$$

Simplify

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Derivative of tan((9x^2-1)/3x)

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The solution