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Identical expressions
y=(log3(4x+ five))/2ctgsqrtx
y equally ( logarithm of 3(4x plus 5)) divide by 2ctg square root of x
y equally ( logarithm of 3(4x plus five)) divide by 2ctg square root of x
y=(log3(4x+5))/2ctg√x
y=log34x+5/2ctgsqrtx
y=(log3(4x+5)) divide by 2ctgsqrtx
Similar expressions
y=(log3(4x-5))/2ctgsqrtx
y=log3(4x+5)/(2ctgsqrtx)
y=(log3(4x+5))/(2ctgsqrtx)

The derivative of the function/ y=(log3(4x+5))/2ctgsqrtx

Derivative of y=(log3(4x+5))/2ctgsqrtx

The solution

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/log(4*x + 5)\           
|------------|           
\   log(3)   /    /  ___\
--------------*cot\\/ x /
      2

$$\frac{\frac{1}{\log{\left(3 \right)}} \log{\left(4 x + 5 \right)}}{2} \cot{\left(\sqrt{x} \right)}$$

((log(4*x + 5)/log(3))/2)*cot(sqrt(x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(4 x + 5 \right)} \cot{\left(\sqrt{x} \right)}$ and $g{\left(x \right)} = 2 \log{\left(3 \right)}$ .

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

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

$\frac{4 \sqrt{x} \sin{\left(2 \sqrt{x} \right)} - 4 x \log{\left(4 x + 5 \right)} - 5 \log{\left(4 x + 5 \right)}}{2 \sqrt{x} \left(1 - \cos{\left(2 \sqrt{x} \right)}\right) \left(4 x + 5\right) \log{\left(3 \right)}}$

The answer is:

$\frac{4 \sqrt{x} \sin{\left(2 \sqrt{x} \right)} - 4 x \log{\left(4 x + 5 \right)} - 5 \log{\left(4 x + 5 \right)}}{2 \sqrt{x} \left(1 - \cos{\left(2 \sqrt{x} \right)}\right) \left(4 x + 5\right) \log{\left(3 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       /  ___\     /        2/  ___\\             
  2*cot\\/ x /     \-1 - cot \\/ x //*log(4*x + 5)
---------------- + -------------------------------
(4*x + 5)*log(3)                ___               
                            4*\/ x *log(3)

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

Simplify

The second derivative [src]

                                                         /            /  ___\\             
                                       /       2/  ___\\ | 1     2*cot\\/ x /|             
                                       \1 + cot \\/ x //*|---- + ------------|*log(5 + 4*x)
       /  ___\     /       2/  ___\\                     | 3/2        x      |             
  8*cot\\/ x /   2*\1 + cot \\/ x //                     \x                  /             
- ------------ - ------------------- + ----------------------------------------------------
            2        ___                                        8                          
   (5 + 4*x)       \/ x *(5 + 4*x)                                                         
-------------------------------------------------------------------------------------------
                                           log(3)

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

Simplify

The third derivative [src]

                                                         /         /       2/  ___\\        2/  ___\        /  ___\\                                    /            /  ___\\
                                       /       2/  ___\\ | 3     2*\1 + cot \\/ x //   4*cot \\/ x /   6*cot\\/ x /|                  /       2/  ___\\ | 1     2*cot\\/ x /|
                                       \1 + cot \\/ x //*|---- + ------------------- + ------------- + ------------|*log(5 + 4*x)   3*\1 + cot \\/ x //*|---- + ------------|
      /  ___\      /       2/  ___\\                     | 5/2            3/2                3/2             2     |                                    | 3/2        x      |
64*cot\\/ x /   12*\1 + cot \\/ x //                     \x              x                  x               x      /                                    \x                  /
------------- + -------------------- - ------------------------------------------------------------------------------------------ + -----------------------------------------
           3        ___          2                                                 16                                                              2*(5 + 4*x)               
  (5 + 4*x)       \/ x *(5 + 4*x)                                                                                                                                            
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                    log(3)

$$\frac{- \frac{\left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right) \left(\frac{6 \cot{\left(\sqrt{x} \right)}}{x^{2}} + \frac{2 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{\frac{3}{2}}} + \frac{4 \cot^{2}{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}}}\right) \log{\left(4 x + 5 \right)}}{16} + \frac{3 \left(\frac{2 \cot{\left(\sqrt{x} \right)}}{x} + \frac{1}{x^{\frac{3}{2}}}\right) \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{2 \left(4 x + 5\right)} + \frac{64 \cot{\left(\sqrt{x} \right)}}{\left(4 x + 5\right)^{3}} + \frac{12 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{\sqrt{x} \left(4 x + 5\right)^{2}}}{\log{\left(3 \right)}}$$

Simplify

The graph

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How to use it?

Identical expressions

Similar expressions

Derivative of y=(log3(4x+5))/2ctgsqrtx

The graph:

Piecewise:

The solution

Method #1

Method #2