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Derivative of 2^x-log3(x-1)

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 x   log(x - 1)
2  - ----------
       log(3)

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

d / x   log(x - 1)\
--|2  - ----------|
dx\       log(3)  /

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

Transform

Detail solution

Differentiate $2^{x} - \frac{\log{\left(x - 1 \right)}}{\log{\left(3 \right)}}$ term by term:
1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $\left(-1\right) 1$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{x - 1}$
    So, the result is: $\frac{1}{\left(x - 1\right) \log{\left(3 \right)}}$
  So, the result is: $- \frac{1}{\left(x - 1\right) \log{\left(3 \right)}}$
The result is: $2^{x} \log{\left(2 \right)} - \frac{1}{\left(x - 1\right) \log{\left(3 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x                1       
2 *log(2) - --------------
            (x - 1)*log(3)

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

Simplify

The second derivative [src]

 x    2             1        
2 *log (2) + ----------------
                     2       
             (-1 + x) *log(3)

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

Simplify

The third derivative [src]

 x    3             2        
2 *log (2) - ----------------
                     3       
             (-1 + x) *log(3)

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

Simplify

The graph