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Identical expressions
z=ln(one - two ^x)
z equally ln(1 minus 2 to the power of x)
z equally ln(one minus two to the power of x)
z=ln(1-2x)
z=ln1-2x
z=ln1-2^x
Similar expressions
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 /

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

log(1 - 2^x)

Detail solution

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

$- \frac{2^{x} \log{\left(2 \right)}}{1 - 2^{x}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

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