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Derivative of (x^2-2*x)/(3+x^2)
Identical expressions
√(x(log(2x)/log(ten)))
√(x( logarithm of (2x) divide by logarithm of (10)))
√(x( logarithm of (2x) divide by logarithm of (ten)))
√xlog2x/log10
√(x(log(2x) divide by log(10)))

The derivative of the function/ √(x(log(2x)/log(10)))

Derivative of √(x(log(2x)/log(10)))

The solution

You have entered [src]

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

$$\sqrt{x \frac{\log{\left(2 x \right)}}{\log{\left(10 \right)}}}$$

sqrt(x*(log(2*x)/log(10)))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    ____________                                
   / x*log(2*x)  /    1        log(2*x)\        
  /  ---------- *|--------- + ---------|*log(10)
\/    log(10)    \2*log(10)   2*log(10)/        
------------------------------------------------
                   x*log(2*x)

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

Simplify

The second derivative [src]

               /                            2                   \
  ____________ |              (1 + log(2*x))    2*(1 + log(2*x))|
\/ x*log(2*x) *|-2*log(2*x) + --------------- - ----------------|
               \                  log(2*x)          log(2*x)    /
-----------------------------------------------------------------
                       2   _________                             
                    4*x *\/ log(10) *log(2*x)

$$\frac{\sqrt{x \log{\left(2 x \right)}} \left(\frac{\left(\log{\left(2 x \right)} + 1\right)^{2}}{\log{\left(2 x \right)}} - \frac{2 \left(\log{\left(2 x \right)} + 1\right)}{\log{\left(2 x \right)}} - 2 \log{\left(2 x \right)}\right)}{4 x^{2} \sqrt{\log{\left(10 \right)}} \log{\left(2 x \right)}}$$

Simplify

The third derivative [src]

               /                                                2                   2                 3                              \
  ____________ |  1      1       1 + log(2*x)   3*(1 + log(2*x))    3*(1 + log(2*x))    (1 + log(2*x))    9*(1 + log(2*x))           |
\/ x*log(2*x) *|- - - -------- + ------------ - ----------------- - ----------------- + --------------- + ---------------- + log(2*x)|
               |  2   log(2*x)       2              4*log(2*x)              2                  2             4*log(2*x)              |
               \                  log (2*x)                            4*log (2*x)        8*log (2*x)                                /
--------------------------------------------------------------------------------------------------------------------------------------
                                                        3   _________                                                                 
                                                       x *\/ log(10) *log(2*x)

$$\frac{\sqrt{x \log{\left(2 x \right)}} \left(\frac{\left(\log{\left(2 x \right)} + 1\right)^{3}}{8 \log{\left(2 x \right)}^{2}} - \frac{3 \left(\log{\left(2 x \right)} + 1\right)^{2}}{4 \log{\left(2 x \right)}} - \frac{3 \left(\log{\left(2 x \right)} + 1\right)^{2}}{4 \log{\left(2 x \right)}^{2}} + \frac{9 \left(\log{\left(2 x \right)} + 1\right)}{4 \log{\left(2 x \right)}} + \frac{\log{\left(2 x \right)} + 1}{\log{\left(2 x \right)}^{2}} + \log{\left(2 x \right)} - \frac{1}{2} - \frac{1}{\log{\left(2 x \right)}}\right)}{x^{3} \sqrt{\log{\left(10 \right)}} \log{\left(2 x \right)}}$$

Simplify

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Derivative of √(x(log(2x)/log(10)))

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The solution