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Derivative of:
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Identical expressions
three ^((one /log(x+ one)))
3 to the power of ((1 divide by logarithm of (x plus 1)))
three to the power of ((one divide by logarithm of (x plus one)))
3((1/log(x+1)))
31/logx+1
3^1/logx+1
3^((1 divide by log(x+1)))
Similar expressions
3^((1/log(x-1)))

The derivative of the function/ 3^((1/log(x+1)))

Derivative of 3^((1/log(x+1)))

The solution

You have entered [src]

     1     
 ----------
 log(x + 1)
3

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

3^(1/log(x + 1))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

     1                                           
 ----------                                      
 log(1 + x) /        2           log(3)  \       
3          *|1 + ---------- + -----------|*log(3)
            |    log(1 + x)      2       |       
            \                 log (1 + x)/       
-------------------------------------------------
                      2    2                     
               (1 + x) *log (1 + x)

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

Simplify

The third derivative [src]

      1                                                                                      
  ---------- /                                    2                                 \        
  log(1 + x) |        6             6          log (3)       3*log(3)      6*log(3) |        
-3          *|2 + ---------- + ----------- + ----------- + ----------- + -----------|*log(3) 
             |    log(1 + x)      2             4             2             3       |        
             \                 log (1 + x)   log (1 + x)   log (1 + x)   log (1 + x)/        
---------------------------------------------------------------------------------------------
                                            3    2                                           
                                     (1 + x) *log (1 + x)

$$- \frac{3^{\frac{1}{\log{\left(x + 1 \right)}}} \left(2 + \frac{6}{\log{\left(x + 1 \right)}} + \frac{3 \log{\left(3 \right)}}{\log{\left(x + 1 \right)}^{2}} + \frac{6}{\log{\left(x + 1 \right)}^{2}} + \frac{6 \log{\left(3 \right)}}{\log{\left(x + 1 \right)}^{3}} + \frac{\log{\left(3 \right)}^{2}}{\log{\left(x + 1 \right)}^{4}}\right) \log{\left(3 \right)}}{\left(x + 1\right)^{3} \log{\left(x + 1 \right)}^{2}}$$

Simplify

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Derivative of 3^((1/log(x+1)))

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The solution