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Identical expressions
log(sqrt(x/ ten ^x))
logarithm of ( square root of (x divide by 10 to the power of x))
logarithm of ( square root of (x divide by ten to the power of x))
log(√(x/10^x))
log(sqrt(x/10x))
logsqrtx/10x
logsqrtx/10^x
log(sqrt(x divide by 10^x))

The derivative of the function/ log(sqrt(x/10^x))

Derivative of log(sqrt(x/10^x))

The solution

You have entered [src]

   /     _____\
   |    /  x  |
log|   /  --- |
   |  /     x |
   \\/    10  /

$$\log{\left(\sqrt{\frac{x}{10^{x}}} \right)}$$

log(sqrt(x/10^x))

Detail solution

Let $u = \sqrt{\frac{x}{10^{x}}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\frac{x}{10^{x}}}$ :
1. Let $u = \frac{x}{10^{x}}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{10^{x}}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = x$ and $g{\left(x \right)} = 10^{x}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. $\frac{d}{d x} 10^{x} = 10^{x} \log{\left(10 \right)}$
    Now plug in to the quotient rule:
    
    $10^{- 2 x} \left(- 10^{x} x \log{\left(10 \right)} + 10^{x}\right)$
  The result of the chain rule is:
  
  $\frac{10^{- \frac{3 x}{2}} \left(- 10^{x} x \log{\left(10 \right)} + 10^{x}\right)}{2 \sqrt{x}}$
The result of the chain rule is:

$\frac{10^{- x} \left(- 10^{x} x \log{\left(10 \right)} + 10^{x}\right)}{2 x}$
Now simplify:

$\frac{- x \log{\left(10 \right)} + 1}{2 x}$

The answer is:

$\frac{- x \log{\left(10 \right)} + 1}{2 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /  -x       -x        \
  x |10     x*10  *log(10)|
10 *|---- - --------------|
    \ 2           2       /
---------------------------
             x

$$\frac{10^{x} \left(- \frac{10^{- x} x \log{\left(10 \right)}}{2} + \frac{10^{- x}}{2}\right)}{x}$$

Simplify

The second derivative [src]

-1 + x*log(10)                                                      
-------------- + (-2 + x*log(10))*log(10) - (-1 + x*log(10))*log(10)
      x                                                             
--------------------------------------------------------------------
                                2*x

$$\frac{\left(x \log{\left(10 \right)} - 2\right) \log{\left(10 \right)} - \left(x \log{\left(10 \right)} - 1\right) \log{\left(10 \right)} + \frac{x \log{\left(10 \right)} - 1}{x}}{2 x}$$

Simplify

The third derivative [src]

                                                2                           2                                                                           
   2                        -1 + x*log(10)   log (10)*(-1 + x*log(10))   log (10)*(-3 + x*log(10))   (-1 + x*log(10))*log(10)   (-2 + x*log(10))*log(10)
log (10)*(-2 + x*log(10)) - -------------- - ------------------------- - ------------------------- + ------------------------ - ------------------------
                                   2                     2                           2                          x                          x            
                                  x                                                                                                                     
--------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                           x

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

Simplify

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Identical expressions

Derivative of log(sqrt(x/10^x))

The graph:

Piecewise:

The solution