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(x^ two + one)/tan(three *x)
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(x2+1)/tan(3*x)
x2+1/tan3*x
(x²+1)/tan(3*x)
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(x^2+1)/tan(3x)
(x2+1)/tan(3x)
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(x^2+1) divide by tan(3*x)
Similar expressions
(x^2-1)/tan(3*x)

The derivative of the function/ (x^2+1)/tan(3*x)

Derivative of (x^2+1)/tan(3*x)

The solution

You have entered [src]

  2     
 x  + 1 
--------
tan(3*x)

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

(x^2 + 1)/tan(3*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} + 1$ and $g{\left(x \right)} = \tan{\left(3 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
2. 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 $\frac{d}{d x} 3 x$ :
    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: $3$
    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 $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $- 3 \sin{\left(3 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           /          2     \ / 2    \
  2*x      \-3 - 3*tan (3*x)/*\x  + 1/
-------- + ---------------------------
tan(3*x)               2              
                    tan (3*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /                                                                                                               /            2     \\
   |                                                                                               /       2     \ |     1 + tan (3*x)||
   |                             /                                   2                    3\   6*x*\1 + tan (3*x)/*|-1 + -------------||
   |         2                   |                    /       2     \      /       2     \ |                       |          2       ||
   |  1 + tan (3*x)     /     2\ |         2        5*\1 + tan (3*x)/    3*\1 + tan (3*x)/ |                       \       tan (3*x)  /|
18*|- ------------- - 3*\1 + x /*|2 + 2*tan (3*x) - ------------------ + ------------------| + ----------------------------------------|
   |       2                     |                         2                    4          |                   tan(3*x)                |
   \    tan (3*x)                \                      tan (3*x)            tan (3*x)     /                                           /

$$18 \left(\frac{6 x \left(\frac{\tan^{2}{\left(3 x \right)} + 1}{\tan^{2}{\left(3 x \right)}} - 1\right) \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan{\left(3 x \right)}} - 3 \left(x^{2} + 1\right) \left(\frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{3}}{\tan^{4}{\left(3 x \right)}} - \frac{5 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2}}{\tan^{2}{\left(3 x \right)}} + 2 \tan^{2}{\left(3 x \right)} + 2\right) - \frac{\tan^{2}{\left(3 x \right)} + 1}{\tan^{2}{\left(3 x \right)}}\right)$$

Simplify

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Derivative of (x^2+1)/tan(3*x)

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The solution