Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 6
Derivative of e^(x*(-3))
Derivative of x-cos(x)
Derivative of x*cos(2*x)
Identical expressions
ctg^ three (log(x+ two , seven))/(x^ five + three *x)
ctg cubed ( logarithm of (x plus 2,7)) divide by (x to the power of 5 plus 3 multiply by x)
ctg to the power of three ( logarithm of (x plus two , seven)) divide by (x to the power of five plus three multiply by x)
ctg3(log(x+2,7))/(x5+3*x)
ctg3logx+2,7/x5+3*x
ctg³(log(x+2,7))/(x⁵+3*x)
ctg to the power of 3(log(x+2,7))/(x to the power of 5+3*x)
ctg^3(log(x+2,7))/(x^5+3x)
ctg3(log(x+2,7))/(x5+3x)
ctg3logx+2,7/x5+3x
ctg^3logx+2,7/x^5+3x
ctg^3(log(x+2,7)) divide by (x^5+3*x)
Similar expressions
ctg^3(log(x+2,7))/(x^5-3*x)
ctg^3(log(x-2,7))/(x^5+3*x)

The derivative of the function/ ctg^3(log(x+2,7))/(x^5+3*x)

Derivative of ctg^3(log(x+2,7))/(x^5+3*x)

The solution

You have entered [src]

   3/   /    27\\
cot |log|x + --||
    \   \    10//
-----------------
      5          
     x  + 3*x

$$\frac{\cot^{3}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{5} + 3 x}$$

cot(log(x + 27/10))^3/(x^5 + 3*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cot^{3}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ and $g{\left(x \right)} = x^{5} + 3 x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ :
  1. There are multiple ways to do this derivative.
    Method #1
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)} = \frac{1}{\tan{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$
    2. Let $u = \tan{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(\log{\left(x + \frac{27}{10} \right)} \right)} = \frac{\sin{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{\cos{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ and $g{\left(x \right)} = \cos{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \log{\left(x + \frac{27}{10} \right)}$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + \frac{27}{10} \right)}$ :
      
      Let $u = x + \frac{27}{10}$ .
      
      The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + \frac{27}{10}\right)$ :
      
      Differentiate $x + \frac{27}{10}$ term by term:
      
      Apply the power rule: $x$ goes to $1$
      
      The derivative of the constant $\frac{27}{10}$ is zero.
      
      The result is: $1$
      
      The result of the chain rule is:
      
      $\frac{1}{x + \frac{27}{10}}$
      
      The result of the chain rule is:
      
      $\frac{\cos{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \log{\left(x + \frac{27}{10} \right)}$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + \frac{27}{10} \right)}$ :
      
      Let $u = x + \frac{27}{10}$ .
      
      The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + \frac{27}{10}\right)$ :
      
      Differentiate $x + \frac{27}{10}$ term by term:
      
      Apply the power rule: $x$ goes to $1$
      
      The derivative of the constant $\frac{27}{10}$ is zero.
      
      The result is: $1$
      
      The result of the chain rule is:
      
      $\frac{1}{x + \frac{27}{10}}$
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}$
      
      Now plug in to the quotient rule:
      
      $\frac{\frac{\sin^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}} + \frac{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}}{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{\frac{\sin^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}} + \frac{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}}{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} \tan^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$
    Method #2
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)} = \frac{\cos{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{\sin{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ and $g{\left(x \right)} = \sin{\left(\log{\left(x + \frac{27}{10} \right)} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \log{\left(x + \frac{27}{10} \right)}$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + \frac{27}{10} \right)}$ :
      
      Let $u = x + \frac{27}{10}$ .
      
      The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + \frac{27}{10}\right)$ :
      
      Differentiate $x + \frac{27}{10}$ term by term:
      
      Apply the power rule: $x$ goes to $1$
      
      The derivative of the constant $\frac{27}{10}$ is zero.
      
      The result is: $1$
      
      The result of the chain rule is:
      
      $\frac{1}{x + \frac{27}{10}}$
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \log{\left(x + \frac{27}{10} \right)}$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + \frac{27}{10} \right)}$ :
      
      Let $u = x + \frac{27}{10}$ .
      
      The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + \frac{27}{10}\right)$ :
      
      Differentiate $x + \frac{27}{10}$ term by term:
      
      Apply the power rule: $x$ goes to $1$
      
      The derivative of the constant $\frac{27}{10}$ is zero.
      
      The result is: $1$
      
      The result of the chain rule is:
      
      $\frac{1}{x + \frac{27}{10}}$
      
      The result of the chain rule is:
      
      $\frac{\cos{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}$
      
      Now plug in to the quotient rule:
      
      $\frac{- \frac{\sin^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}} - \frac{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}}{\sin^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$
  The result of the chain rule is:
  
  $- \frac{3 \left(\frac{\sin^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}} + \frac{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}\right) \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} \tan^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{5} + 3 x$ term by term:
  1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
  2. 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 is: $5 x^{4} + 3$
Now plug in to the quotient rule:

$\frac{- \left(5 x^{4} + 3\right) \cot^{3}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} - \frac{3 \left(x^{5} + 3 x\right) \left(\frac{\sin^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}} + \frac{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x + \frac{27}{10}}\right) \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{\cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} \tan^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}}{\left(x^{5} + 3 x\right)^{2}}$
Now simplify:

$- \frac{\left(30 x \left(x^{4} + 3\right) + \frac{\left(10 x + 27\right) \left(5 x^{4} + 3\right) \sin{\left(2 \log{\left(x + \frac{27}{10} \right)} \right)}}{2}\right) \cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{2} \left(10 x + 27\right) \left(x^{4} + 3\right)^{2} \sin^{4}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$

The answer is:

$- \frac{\left(30 x \left(x^{4} + 3\right) + \frac{\left(10 x + 27\right) \left(5 x^{4} + 3\right) \sin{\left(2 \log{\left(x + \frac{27}{10} \right)} \right)}}{2}\right) \cos^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{2} \left(10 x + 27\right) \left(x^{4} + 3\right)^{2} \sin^{4}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3/   /    27\\ /        4\        2/   /    27\\ /        2/   /    27\\\
cot |log|x + --||*\-3 - 5*x /   3*cot |log|x + --||*|-1 - cot |log|x + --|||
    \   \    10//                     \   \    10// \         \   \    10///
----------------------------- + --------------------------------------------
                   2                        /    27\ / 5      \             
         / 5      \                         |x + --|*\x  + 3*x/             
         \x  + 3*x/                         \    10/

$$\frac{\left(- 5 x^{4} - 3\right) \cot^{3}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{\left(x^{5} + 3 x\right)^{2}} + \frac{3 \left(- \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} - 1\right) \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{\left(x + \frac{27}{10}\right) \left(x^{5} + 3 x\right)}$$

Simplify

The second derivative [src]

  /                    /                 2\                                                                                                                                    \                 
  |                    |       /       4\ |                                                                                                                                    |                 
  |     2/   /27    \\ |       \3 + 5*x / |                                                                                                                                    |                 
  |  cot |log|-- + x||*|10*x - -----------|       /       2/   /27    \\\ /         2/   /27    \\      /   /27    \\\      /       2/   /27    \\\ /       4\    /   /27    \\|                 
  |      \   \10    // |        3 /     4\|   150*|1 + cot |log|-- + x|||*|2 + 4*cot |log|-- + x|| + cot|log|-- + x|||   30*|1 + cot |log|-- + x|||*\3 + 5*x /*cot|log|-- + x|||                 
  |                    \       x *\3 + x //       \        \   \10    /// \          \   \10    //      \   \10    ///      \        \   \10    ///               \   \10    //|    /   /27    \\
2*|- -------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------|*cot|log|-- + x||
  |                       4                                                             2                                                2 /     4\                            |    \   \10    //
  \                  3 + x                                                 x*(27 + 10*x)                                                x *\3 + x /*(27 + 10*x)                /                 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                   4                                                                                             
                                                                                              3 + x

$$\frac{2 \left(- \frac{\left(10 x - \frac{\left(5 x^{4} + 3\right)^{2}}{x^{3} \left(x^{4} + 3\right)}\right) \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{4} + 3} + \frac{150 \left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right) \left(4 \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 2\right)}{x \left(10 x + 27\right)^{2}} + \frac{30 \left(5 x^{4} + 3\right) \left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right) \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{2} \left(10 x + 27\right) \left(x^{4} + 3\right)}\right) \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{4} + 3}$$

Simplify

The third derivative [src]

  /                    /                               3 \                                                                                                                                                                                                                                                                                                                                                                                                \
  |                    |        /       4\   /       4\  |                                                                                                                                                                                                                                                                     /                 2\                                                                                                       |
  |     3/   /27    \\ |     20*\3 + 5*x /   \3 + 5*x /  |                                                                                                                                                                                                                                                                     |       /       4\ |                                                                                                       |
  |  cot |log|-- + x||*|10 - ------------- + ------------|                                /                       2                                                                                                                                                           \         2/   /27    \\ /       2/   /27    \\\ |       \3 + 5*x / |                                                                                                       |
  |      \   \10    // |              4                 2|        /       2/   /27    \\\ |/       2/   /27    \\\       2/   /27    \\        4/   /27    \\        3/   /27    \\     /       2/   /27    \\\    /   /27    \\        2/   /27    \\ /       2/   /27    \\\|   30*cot |log|-- + x||*|1 + cot |log|-- + x|||*|10*x - -----------|       /       2/   /27    \\\ /       4\ /         2/   /27    \\      /   /27    \\\    /   /27    \\|
  |                    |         3 + x        4 /     4\ |   1000*|1 + cot |log|-- + x|||*||1 + cot |log|-- + x|||  + cot |log|-- + x|| + 2*cot |log|-- + x|| + 3*cot |log|-- + x|| + 3*|1 + cot |log|-- + x|||*cot|log|-- + x|| + 7*cot |log|-- + x||*|1 + cot |log|-- + x||||          \   \10    // \        \   \10    /// |        3 /     4\|   150*|1 + cot |log|-- + x|||*\3 + 5*x /*|2 + 4*cot |log|-- + x|| + cot|log|-- + x|||*cot|log|-- + x|||
  |                    \                     x *\3 + x / /        \        \   \10    /// \\        \   \10    ///        \   \10    //         \   \10    //         \   \10    //     \        \   \10    ///    \   \10    //         \   \10    // \        \   \10    ////                                                \       x *\3 + x //       \        \   \10    ///            \          \   \10    //      \   \10    ///    \   \10    //|
6*|- ----------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ----------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------|
  |                               4                                                                                                                                         3                                                                                                                            /     4\                                                                            2 /     4\            2                                      |
  \                          3 + x                                                                                                                             x*(27 + 10*x)                                                                                                                             \3 + x /*(27 + 10*x)                                                               x *\3 + x /*(27 + 10*x)                                       /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                4                                                                                                                                                                                                                          
                                                                                                                                                                                                                           3 + x

$$\frac{6 \left(- \frac{\left(10 - \frac{20 \left(5 x^{4} + 3\right)}{x^{4} + 3} + \frac{\left(5 x^{4} + 3\right)^{3}}{x^{4} \left(x^{4} + 3\right)^{2}}\right) \cot^{3}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{4} + 3} + \frac{30 \left(10 x - \frac{\left(5 x^{4} + 3\right)^{2}}{x^{3} \left(x^{4} + 3\right)}\right) \left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right) \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{\left(10 x + 27\right) \left(x^{4} + 3\right)} - \frac{1000 \left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right) \left(\left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right)^{2} + 7 \left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right) \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 3 \left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right) \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 2 \cot^{4}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 3 \cot^{3}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)}\right)}{x \left(10 x + 27\right)^{3}} - \frac{150 \left(5 x^{4} + 3\right) \left(\cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 1\right) \left(4 \cot^{2}{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)} + 2\right) \cot{\left(\log{\left(x + \frac{27}{10} \right)} \right)}}{x^{2} \left(10 x + 27\right)^{2} \left(x^{4} + 3\right)}\right)}{x^{4} + 3}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ctg^3(log(x+2,7))/(x^5+3*x)

The graph:

Piecewise:

The solution

Method #1

Method #2