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Identical expressions
log_4(eight ^x+ two ^x)
logarithm of _4(8 to the power of x plus 2 to the power of x)
logarithm of _4(eight to the power of x plus two to the power of x)
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Similar expressions
log_4(8^x-2^x)

The derivative of the function/ log_4(8^x+2^x)

Derivative of log_4(8^x+2^x)

The solution

You have entered [src]

   / x    x\
log\8  + 2 /
------------
   log(4)

$$\frac{\log{\left(2^{x} + 8^{x} \right)}}{\log{\left(4 \right)}}$$

log(8^x + 2^x)/log(4)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = 2^{x} + 8^{x}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2^{x} + 8^{x}\right)$ :
  1. Differentiate $2^{x} + 8^{x}$ term by term:
    1. $\frac{d}{d x} 8^{x} = 8^{x} \log{\left(8 \right)}$
    2. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
    The result is: $2^{x} \log{\left(2 \right)} + 8^{x} \log{\left(8 \right)}$
  The result of the chain rule is:
  
  $\frac{2^{x} \log{\left(2 \right)} + 8^{x} \log{\left(8 \right)}}{2^{x} + 8^{x}}$
So, the result is: $\frac{2^{x} \log{\left(2 \right)} + 8^{x} \log{\left(8 \right)}}{\left(2^{x} + 8^{x}\right) \log{\left(4 \right)}}$
Now simplify:

$\frac{\frac{2^{x}}{2} + \frac{3 \cdot 8^{x}}{2}}{2^{x} + 8^{x}}$

The answer is:

$\frac{\frac{2^{x}}{2} + \frac{3 \cdot 8^{x}}{2}}{2^{x} + 8^{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x           x       
2 *log(2) + 8 *log(8)
---------------------
   / x    x\         
   \8  + 2 /*log(4)

$$\frac{2^{x} \log{\left(2 \right)} + 8^{x} \log{\left(8 \right)}}{\left(2^{x} + 8^{x}\right) \log{\left(4 \right)}}$$

Simplify

The second derivative [src]

                                                 2
                          / x           x       \ 
 x    2       x    2      \2 *log(2) + 8 *log(8)/ 
2 *log (2) + 8 *log (8) - ------------------------
                                   x    x         
                                  2  + 8          
--------------------------------------------------
                 / x    x\                        
                 \2  + 8 /*log(4)

$$\frac{2^{x} \log{\left(2 \right)}^{2} + 8^{x} \log{\left(8 \right)}^{2} - \frac{\left(2^{x} \log{\left(2 \right)} + 8^{x} \log{\left(8 \right)}\right)^{2}}{2^{x} + 8^{x}}}{\left(2^{x} + 8^{x}\right) \log{\left(4 \right)}}$$

Simplify

The third derivative [src]

                                                   3                                                      
                            / x           x       \      / x    2       x    2   \ / x           x       \
 x    3       x    3      2*\2 *log(2) + 8 *log(8)/    3*\2 *log (2) + 8 *log (8)/*\2 *log(2) + 8 *log(8)/
2 *log (2) + 8 *log (8) + -------------------------- - ---------------------------------------------------
                                           2                                  x    x                      
                                  / x    x\                                  2  + 8                       
                                  \2  + 8 /                                                               
----------------------------------------------------------------------------------------------------------
                                             / x    x\                                                    
                                             \2  + 8 /*log(4)

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

Simplify

The graph

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Derivative of log_4(8^x+2^x)

The graph:

Piecewise:

The solution