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

The derivative of the function/ y=log(3)(4x+5)/(2ctgsqrtx)

Derivative of y=log(3)(4x+5)/(2ctgsqrtx)

The solution

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Differentiate $4 x + 5$ term by term:
    1. The derivative of the constant $5$ is zero.
    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: $4$
    The result is: $4$
  So, the result is: $4 \log{\left(3 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
      
      $\frac{- \frac{\sin^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}} - \frac{\cos^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}}}{\sin^{2}{\left(\sqrt{x} \right)}}$
  So, the result is: $- \frac{2 \left(\frac{\sin^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}} + \frac{\cos^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}}\right)}{\cos^{2}{\left(\sqrt{x} \right)} \tan^{2}{\left(\sqrt{x} \right)}}$
Now plug in to the quotient rule:

$\frac{\frac{2 \left(4 x + 5\right) \left(\frac{\sin^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}} + \frac{\cos^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}}\right) \log{\left(3 \right)}}{\cos^{2}{\left(\sqrt{x} \right)} \tan^{2}{\left(\sqrt{x} \right)}} + 8 \log{\left(3 \right)} \cot{\left(\sqrt{x} \right)}}{4 \cot^{2}{\left(\sqrt{x} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                  /                                                                                                                             /                        /       2/  ___\\\\       
                  |                                                                                                                             |2          1          2*\1 + cot \\/ x //||       
                  |          /                                                                                                          2\   24*|- + --------------- - -------------------||       
                  |          |                                             /       2/  ___\\     /       2/  ___\\     /       2/  ___\\ |      |x    3/2    /  ___\           2/  ___\   ||       
/       2/  ___\\ |          | 4            3                 6         10*\1 + cot \\/ x //   6*\1 + cot \\/ x //   6*\1 + cot \\/ x // |      \    x   *cot\\/ x /      x*cot \\/ x /   /|       
\1 + cot \\/ x //*|(5 + 4*x)*|---- + ---------------- + ------------- - -------------------- - ------------------- + --------------------| - ----------------------------------------------|*log(3)
                  |          | 3/2    5/2    2/  ___\    2    /  ___\      3/2    2/  ___\         2    3/  ___\        3/2    4/  ___\  |                        /  ___\                  |       
                  \          \x      x   *cot \\/ x /   x *cot\\/ x /     x   *cot \\/ x /        x *cot \\/ x /       x   *cot \\/ x /  /                     cot\\/ x /                  /       
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                 16

$$\frac{\left(\left(4 x + 5\right) \left(- \frac{6 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{2} \cot^{3}{\left(\sqrt{x} \right)}} + \frac{6}{x^{2} \cot{\left(\sqrt{x} \right)}} + \frac{6 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)^{2}}{x^{\frac{3}{2}} \cot^{4}{\left(\sqrt{x} \right)}} - \frac{10 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{\frac{3}{2}} \cot^{2}{\left(\sqrt{x} \right)}} + \frac{4}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}} \cot^{2}{\left(\sqrt{x} \right)}}\right) - \frac{24 \left(- \frac{2 \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right)}{x \cot^{2}{\left(\sqrt{x} \right)}} + \frac{2}{x} + \frac{1}{x^{\frac{3}{2}} \cot{\left(\sqrt{x} \right)}}\right)}{\cot{\left(\sqrt{x} \right)}}\right) \left(\cot^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(3 \right)}}{16}$$

Simplify

The graph

Derivative of y=log(3)(4x+5)/(2ctgsqrtx)

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How to use it?

Identical expressions

Similar expressions

Derivative of y=log(3)(4x+5)/(2ctgsqrtx)

The graph:

Piecewise:

The solution

Method #1

Method #2