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z=ln(1-2^x)

The derivative of the function/ z=ln(1+2^x)

Derivative of z=ln(1+2^x)

The solution

You have entered [src]

   /     x\
log\1 + 2 /

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

log(1 + 2^x)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of z=ln(1+2^x)

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The solution