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y=ln(2^x-3^x)

The derivative of the function/ y=ln(2^x+3^x)

Derivative of y=ln(2^x+3^x)

The solution

You have entered [src]

   / x    x\
log\2  + 3 /

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

log(2^x + 3^x)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=ln(2^x+3^x)

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The solution