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Derivative of:
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Derivative of (x-4)
Identical expressions
(log(x- four *sqrt(x- two)+ five)/log(sqrt(three)))
( logarithm of (x minus 4 multiply by square root of (x minus 2) plus 5) divide by logarithm of ( square root of (3)))
( logarithm of (x minus four multiply by square root of (x minus two) plus five) divide by logarithm of ( square root of (three)))
(log(x-4*√(x-2)+5)/log(√(3)))
(log(x-4sqrt(x-2)+5)/log(sqrt(3)))
logx-4sqrtx-2+5/logsqrt3
(log(x-4*sqrt(x-2)+5) divide by log(sqrt(3)))
Similar expressions
(log(x-4*sqrt(x-2)-5)/log(sqrt(3)))
(log(x+4*sqrt(x-2)+5)/log(sqrt(3)))
(log(x-4*sqrt(x+2)+5)/log(sqrt(3)))

The derivative of the function/ (log(x-4*sqrt(x-2)+5)/log(sqrt(3)))

Derivative of (log(x-4*sqrt(x-2)+5)/log(sqrt(3)))

The solution

You have entered [src]

   /        _______    \
log\x - 4*\/ x - 2  + 5/
------------------------
          /  ___\       
       log\\/ 3 /

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \left(x - 4 \sqrt{x - 2}\right) + 5$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x - 4 \sqrt{x - 2}\right) + 5\right)$ :
  1. Differentiate $\left(x - 4 \sqrt{x - 2}\right) + 5$ term by term:
    1. Differentiate $x - 4 \sqrt{x - 2}$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Let $u = x - 2$ .
        
        Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
        
        Differentiate $x - 2$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $-2$ is zero.
        
        The result is: $1$
        
        The result of the chain rule is:
        
        $\frac{1}{2 \sqrt{x - 2}}$
        
        So, the result is: $- \frac{2}{\sqrt{x - 2}}$
      The result is: $1 - \frac{2}{\sqrt{x - 2}}$
    2. The derivative of the constant $5$ is zero.
    The result is: $1 - \frac{2}{\sqrt{x - 2}}$
  The result of the chain rule is:
  
  $\frac{1 - \frac{2}{\sqrt{x - 2}}}{\left(x - 4 \sqrt{x - 2}\right) + 5}$
So, the result is: $\frac{1 - \frac{2}{\sqrt{x - 2}}}{\left(\left(x - 4 \sqrt{x - 2}\right) + 5\right) \log{\left(\sqrt{3} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 2              
         1 - ---------          
               _______          
             \/ x - 2           
--------------------------------
/        _______    \    /  ___\
\x - 4*\/ x - 2  + 5/*log\\/ 3 /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of (log(x-4*sqrt(x-2)+5)/log(sqrt(3)))

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (log(x-4*sqrt(x-2)+5)/log(sqrt(3)))

The graph:

Piecewise:

The solution