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Identical expressions
three ^(log(x+ one)/log(five))
3 to the power of ( logarithm of (x plus 1) divide by logarithm of (5))
three to the power of ( logarithm of (x plus one) divide by logarithm of (five))
3(log(x+1)/log(5))
3logx+1/log5
3^logx+1/log5
3^(log(x+1) divide by log(5))
Similar expressions
3^(log(x-1)/log(5))

The derivative of the function/ 3^(log(x+1)/log(5))

Derivative of 3^(log(x+1)/log(5))

The solution

You have entered [src]

 log(x + 1)
 ----------
   log(5)  
3

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

3^(log(x + 1)/log(5))

Detail solution

Let $u = \frac{\log{\left(x + 1 \right)}}{\log{\left(5 \right)}}$ .
$\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(x + 1 \right)}}{\log{\left(5 \right)}}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\frac{1}{x + 1}$
  So, the result is: $\frac{1}{\left(x + 1\right) \log{\left(5 \right)}}$
The result of the chain rule is:

$\frac{3^{\frac{\log{\left(x + 1 \right)}}{\log{\left(5 \right)}}} \log{\left(3 \right)}}{\left(x + 1\right) \log{\left(5 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 log(x + 1)       
 ----------       
   log(5)         
3          *log(3)
------------------
  (x + 1)*log(5)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 3^(log(x+1)/log(5))

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The solution