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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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
  1. Let .

  2. Then, apply the chain rule. Multiply by :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. The derivative of is .

      3. Then, apply the chain rule. Multiply by :

        1. Differentiate term by term:

          1. Apply the power rule: goes to

          2. The derivative of the constant is zero.

          The result is:

        The result of the chain rule is:

      The result of the chain rule is:

    The result of the chain rule is:

  3. Now simplify:


The answer is:

The graph
The first derivative [src]
      1             
  ----------        
  log(x + 1)        
-3          *log(3) 
--------------------
           2        
(x + 1)*log (x + 1) 
$$- \frac{3^{\frac{1}{\log{\left(x + 1 \right)}}} \log{\left(3 \right)}}{\left(x + 1\right) \log{\left(x + 1 \right)}^{2}}$$
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)              
$$\frac{3^{\frac{1}{\log{\left(x + 1 \right)}}} \left(1 + \frac{2}{\log{\left(x + 1 \right)}} + \frac{\log{\left(3 \right)}}{\log{\left(x + 1 \right)}^{2}}\right) \log{\left(3 \right)}}{\left(x + 1\right)^{2} \log{\left(x + 1 \right)}^{2}}$$
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)                                    
$$- \frac{3^{\frac{1}{\log{\left(x + 1 \right)}}} \left(2 + \frac{6}{\log{\left(x + 1 \right)}} + \frac{3 \log{\left(3 \right)}}{\log{\left(x + 1 \right)}^{2}} + \frac{6}{\log{\left(x + 1 \right)}^{2}} + \frac{6 \log{\left(3 \right)}}{\log{\left(x + 1 \right)}^{3}} + \frac{\log{\left(3 \right)}^{2}}{\log{\left(x + 1 \right)}^{4}}\right) \log{\left(3 \right)}}{\left(x + 1\right)^{3} \log{\left(x + 1 \right)}^{2}}$$