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Derivative of 1/x^10
Derivative of e^(x*(-2))
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Identical expressions
sqrt((two *x+ three)/(two *x- three))*ctg(three *x^ two + five)
square root of ((2 multiply by x plus 3) divide by (2 multiply by x minus 3)) multiply by ctg(3 multiply by x squared plus 5)
square root of ((two multiply by x plus three) divide by (two multiply by x minus three)) multiply by ctg(three multiply by x to the power of two plus five)
√((2*x+3)/(2*x-3))*ctg(3*x^2+5)
sqrt((2*x+3)/(2*x-3))*ctg(3*x2+5)
sqrt2*x+3/2*x-3*ctg3*x2+5
sqrt((2*x+3)/(2*x-3))*ctg(3*x²+5)
sqrt((2*x+3)/(2*x-3))*ctg(3*x to the power of 2+5)
sqrt((2x+3)/(2x-3))ctg(3x^2+5)
sqrt((2x+3)/(2x-3))ctg(3x2+5)
sqrt2x+3/2x-3ctg3x2+5
sqrt2x+3/2x-3ctg3x^2+5
sqrt((2*x+3) divide by (2*x-3))*ctg(3*x^2+5)
Similar expressions
sqrt((2*x+3)/(2*x-3))*ctg(3*x^2-5)
sqrt((2*x-3)/(2*x-3))*ctg(3*x^2+5)
sqrt((2*x+3)/(2*x+3))*ctg(3*x^2+5)

The derivative of the function/ sqrt((2*x+3)/(2*x-3))*ctg(3*x^2+5)

Derivative of sqrt((2x+3)/(2x-3))ctg(3x^2+5)

The solution

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    _________              
   / 2*x + 3     /   2    \
  /  ------- *cot\3*x  + 5/
\/   2*x - 3

$$\sqrt{\frac{2 x + 3}{2 x - 3}} \cot{\left(3 x^{2} + 5 \right)}$$

sqrt((2*x + 3)/(2*x - 3))*cot(3*x^2 + 5)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{\frac{2 x + 3}{2 x - 3}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{2 x + 3}{2 x - 3}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 3}{2 x - 3}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 2 x + 3$ and $g{\left(x \right)} = 2 x - 3$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $2 x + 3$ term by term:
      1. The derivative of the constant $3$ is zero.
      2. 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 is: $2$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Differentiate $2 x - 3$ term by term:
      1. The derivative of the constant $-3$ is zero.
      2. 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 is: $2$
    Now plug in to the quotient rule:
    
    $- \frac{12}{\left(2 x - 3\right)^{2}}$
  The result of the chain rule is:
  
  $- \frac{6}{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right)^{2}}$
$g{\left(x \right)} = \cot{\left(3 x^{2} + 5 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(3 x^{2} + 5 \right)} = \frac{1}{\tan{\left(3 x^{2} + 5 \right)}}$
  2. Let $u = \tan{\left(3 x^{2} + 5 \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 x^{2} + 5 \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(3 x^{2} + 5 \right)} = \frac{\sin{\left(3 x^{2} + 5 \right)}}{\cos{\left(3 x^{2} + 5 \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(3 x^{2} + 5 \right)}$ and $g{\left(x \right)} = \cos{\left(3 x^{2} + 5 \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 3 x^{2} + 5$ .
      
      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 $\frac{d}{d x} \left(3 x^{2} + 5\right)$ :
      
      Differentiate $3 x^{2} + 5$ term by term:
      
      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: $6 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $6 x$
      
      The result of the chain rule is:
      
      $6 x \cos{\left(3 x^{2} + 5 \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 3 x^{2} + 5$ .
      
      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 $\frac{d}{d x} \left(3 x^{2} + 5\right)$ :
      
      Differentiate $3 x^{2} + 5$ term by term:
      
      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: $6 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $6 x$
      
      The result of the chain rule is:
      
      $- 6 x \sin{\left(3 x^{2} + 5 \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{6 x \sin^{2}{\left(3 x^{2} + 5 \right)} + 6 x \cos^{2}{\left(3 x^{2} + 5 \right)}}{\cos^{2}{\left(3 x^{2} + 5 \right)}}$
    The result of the chain rule is:
    
    $- \frac{6 x \sin^{2}{\left(3 x^{2} + 5 \right)} + 6 x \cos^{2}{\left(3 x^{2} + 5 \right)}}{\cos^{2}{\left(3 x^{2} + 5 \right)} \tan^{2}{\left(3 x^{2} + 5 \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(3 x^{2} + 5 \right)} = \frac{\cos{\left(3 x^{2} + 5 \right)}}{\sin{\left(3 x^{2} + 5 \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(3 x^{2} + 5 \right)}$ and $g{\left(x \right)} = \sin{\left(3 x^{2} + 5 \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 3 x^{2} + 5$ .
    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 $\frac{d}{d x} \left(3 x^{2} + 5\right)$ :
      
      Differentiate $3 x^{2} + 5$ term by term:
      
      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: $6 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $6 x$
      
      The result of the chain rule is:
      
      $- 6 x \sin{\left(3 x^{2} + 5 \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 3 x^{2} + 5$ .
    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 $\frac{d}{d x} \left(3 x^{2} + 5\right)$ :
      
      Differentiate $3 x^{2} + 5$ term by term:
      
      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: $6 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $6 x$
      
      The result of the chain rule is:
      
      $6 x \cos{\left(3 x^{2} + 5 \right)}$
    Now plug in to the quotient rule:
    
    $\frac{- 6 x \sin^{2}{\left(3 x^{2} + 5 \right)} - 6 x \cos^{2}{\left(3 x^{2} + 5 \right)}}{\sin^{2}{\left(3 x^{2} + 5 \right)}}$
The result is: $- \frac{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(6 x \sin^{2}{\left(3 x^{2} + 5 \right)} + 6 x \cos^{2}{\left(3 x^{2} + 5 \right)}\right)}{\cos^{2}{\left(3 x^{2} + 5 \right)} \tan^{2}{\left(3 x^{2} + 5 \right)}} - \frac{6 \cot{\left(3 x^{2} + 5 \right)}}{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right)^{2}}$
Now simplify:

$- \frac{6 x \left(2 x - 3\right) \left(2 x + 3\right) + 6 \sin{\left(3 x^{2} + 5 \right)} \cos{\left(3 x^{2} + 5 \right)}}{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right)^{2} \sin^{2}{\left(3 x^{2} + 5 \right)}}$

The answer is:

$- \frac{6 x \left(2 x - 3\right) \left(2 x + 3\right) + 6 \sin{\left(3 x^{2} + 5 \right)} \cos{\left(3 x^{2} + 5 \right)}}{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right)^{2} \sin^{2}{\left(3 x^{2} + 5 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                              _________                                               
                                             / 2*x + 3  /   1       2*x + 3  \              /   2    \
                                            /  ------- *|------- - ----------|*(2*x - 3)*cot\3*x  + 5/
        _________                         \/   2*x - 3  |2*x - 3            2|                        
       / 2*x + 3  /        2/   2    \\                 \          (2*x - 3) /                        
6*x*  /  ------- *\-1 - cot \3*x  + 5// + ------------------------------------------------------------
    \/   2*x - 3                                                    2*x + 3

$$6 x \sqrt{\frac{2 x + 3}{2 x - 3}} \left(- \cot^{2}{\left(3 x^{2} + 5 \right)} - 1\right) + \frac{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right) \left(\frac{1}{2 x - 3} - \frac{2 x + 3}{\left(2 x - 3\right)^{2}}\right) \cot{\left(3 x^{2} + 5 \right)}}{2 x + 3}$$

Simplify

The second derivative [src]

                /                                                                                 /                         3 + 2*x \                                                         \
                |                                                                                 |                     1 - --------|                                                         |
                |                                                                  /    3 + 2*x \ |   2          2          -3 + 2*x|    /       2\        /       2/       2\\ /    3 + 2*x \|
     __________ |                                                                  |1 - --------|*|-------- + ------- - ------------|*cot\5 + 3*x /   12*x*\1 + cot \5 + 3*x //*|1 - --------||
    / 3 + 2*x   |         2/       2\       2 /       2/       2\\    /       2\   \    -3 + 2*x/ \-3 + 2*x   3 + 2*x     3 + 2*x   /                                           \    -3 + 2*x/|
-  /  -------- *|6 + 6*cot \5 + 3*x / - 72*x *\1 + cot \5 + 3*x //*cot\5 + 3*x / + ---------------------------------------------------------------- + ----------------------------------------|
 \/   -3 + 2*x  \                                                                                              3 + 2*x                                                3 + 2*x                 /

$$- \sqrt{\frac{2 x + 3}{2 x - 3}} \left(- 72 x^{2} \left(\cot^{2}{\left(3 x^{2} + 5 \right)} + 1\right) \cot{\left(3 x^{2} + 5 \right)} + \frac{12 x \left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(\cot^{2}{\left(3 x^{2} + 5 \right)} + 1\right)}{2 x + 3} + \frac{\left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(- \frac{1 - \frac{2 x + 3}{2 x - 3}}{2 x + 3} + \frac{2}{2 x + 3} + \frac{2}{2 x - 3}\right) \cot{\left(3 x^{2} + 5 \right)}}{2 x + 3} + 6 \cot^{2}{\left(3 x^{2} + 5 \right)} + 6\right)$$

Simplify

The third derivative [src]

               /                                                                                                                                                                                                      /                                         2                                                                 \                                                                                             \
               |                                                                                                                                                                                                      |                           /    3 + 2*x \      /    3 + 2*x \                              /    3 + 2*x \  |                                                                                             |
               |                                                                                                                                                                                                      |                           |1 - --------|    6*|1 - --------|                            6*|1 - --------|  |                                                          /                         3 + 2*x \|
               |                                                                                                                                                                                       /    3 + 2*x \ |     8            8        \    -3 + 2*x/      \    -3 + 2*x/            8                 \    -3 + 2*x/  |    /       2\                                            |                     1 - --------||
               |                                                                                                      /    3 + 2*x \ /       2/       2\       2 /       2/       2\\    /       2\\   |1 - --------|*|----------- + ---------- + --------------- - ---------------- + -------------------- - --------------------|*cot\5 + 3*x /        /       2/       2\\ /    3 + 2*x \ |   2          2          -3 + 2*x||
    __________ |                                                                                                   18*|1 - --------|*\1 + cot \5 + 3*x / - 12*x *\1 + cot \5 + 3*x //*cot\5 + 3*x //   \    -3 + 2*x/ |          2            2               2                 2      (-3 + 2*x)*(3 + 2*x)   (-3 + 2*x)*(3 + 2*x)|                 18*x*\1 + cot \5 + 3*x //*|1 - --------|*|-------- + ------- - ------------||
   / 3 + 2*x   |        /       2/       2\\ /     /       2\      2 /       2/       2\\      2    2/       2\\      \    -3 + 2*x/                                                                                  \(-3 + 2*x)    (3 + 2*x)       (3 + 2*x)         (3 + 2*x)                                                  /                                           \    -3 + 2*x/ \-3 + 2*x   3 + 2*x     3 + 2*x   /|
  /  -------- *|- 216*x*\1 + cot \5 + 3*x //*\- cot\5 + 3*x / + 2*x *\1 + cot \5 + 3*x // + 4*x *cot \5 + 3*x // - --------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------ + ----------------------------------------------------------------------------|
\/   -3 + 2*x  \                                                                                                                                        3 + 2*x                                                                                                         3 + 2*x                                                                                                       3 + 2*x                                   /

$$\sqrt{\frac{2 x + 3}{2 x - 3}} \left(\frac{18 x \left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(\cot^{2}{\left(3 x^{2} + 5 \right)} + 1\right) \left(- \frac{1 - \frac{2 x + 3}{2 x - 3}}{2 x + 3} + \frac{2}{2 x + 3} + \frac{2}{2 x - 3}\right)}{2 x + 3} - 216 x \left(\cot^{2}{\left(3 x^{2} + 5 \right)} + 1\right) \left(2 x^{2} \left(\cot^{2}{\left(3 x^{2} + 5 \right)} + 1\right) + 4 x^{2} \cot^{2}{\left(3 x^{2} + 5 \right)} - \cot{\left(3 x^{2} + 5 \right)}\right) - \frac{18 \left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(- 12 x^{2} \left(\cot^{2}{\left(3 x^{2} + 5 \right)} + 1\right) \cot{\left(3 x^{2} + 5 \right)} + \cot^{2}{\left(3 x^{2} + 5 \right)} + 1\right)}{2 x + 3} + \frac{\left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(\frac{\left(1 - \frac{2 x + 3}{2 x - 3}\right)^{2}}{\left(2 x + 3\right)^{2}} - \frac{6 \left(1 - \frac{2 x + 3}{2 x - 3}\right)}{\left(2 x + 3\right)^{2}} - \frac{6 \left(1 - \frac{2 x + 3}{2 x - 3}\right)}{\left(2 x - 3\right) \left(2 x + 3\right)} + \frac{8}{\left(2 x + 3\right)^{2}} + \frac{8}{\left(2 x - 3\right) \left(2 x + 3\right)} + \frac{8}{\left(2 x - 3\right)^{2}}\right) \cot{\left(3 x^{2} + 5 \right)}}{2 x + 3}\right)$$

Simplify

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Derivative of sqrt((2x+3)/(2x-3))ctg(3x^2+5)

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt((2*x+3)/(2*x-3))*ctg(3*x^2+5)

The graph:

Piecewise:

The solution

Method #1

Method #2

Derivative of sqrt((2x+3)/(2x-3))ctg(3x^2+5)