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(three ^x-ln(x))/(x- four)
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(3x-ln(x))/(x-4)
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(3^x-ln(x)) divide by (x-4)
Similar expressions
(3^x-ln(x))/(x+4)
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The derivative of the function/ (3^x-ln(x))/(x-4)

Derivative of (3^x-ln(x))/(x-4)

The solution

You have entered [src]

 x         
3  - log(x)
-----------
   x - 4

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

(3^x - log(x))/(x - 4)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 3^{x} - \log{\left(x \right)}$ and $g{\left(x \right)} = x - 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3^{x} - \log{\left(x \right)}$ term by term:
  1. $\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    So, the result is: $- \frac{1}{x}$
  The result is: $3^{x} \log{\left(3 \right)} - \frac{1}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x - 4$ term by term:
  1. The derivative of the constant $-4$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{- 3^{x} + \left(x - 4\right) \left(3^{x} \log{\left(3 \right)} - \frac{1}{x}\right) + \log{\left(x \right)}}{\left(x - 4\right)^{2}}$
Now simplify:

$\frac{x \left(- 3^{x} + \log{\left(x \right)}\right) + \left(x - 4\right) \left(3^{x} x \log{\left(3 \right)} - 1\right)}{x \left(x - 4\right)^{2}}$

The answer is:

$\frac{x \left(- 3^{x} + \log{\left(x \right)}\right) + \left(x - 4\right) \left(3^{x} x \log{\left(3 \right)} - 1\right)}{x \left(x - 4\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1    x                     
- - + 3 *log(3)    x         
  x               3  - log(x)
--------------- - -----------
     x - 4                 2 
                    (x - 4)

$$- \frac{3^{x} - \log{\left(x \right)}}{\left(x - 4\right)^{2}} + \frac{3^{x} \log{\left(3 \right)} - \frac{1}{x}}{x - 4}$$

Simplify

The second derivative [src]

                    /  1    x       \                  
                  2*|- - + 3 *log(3)|     / x         \
1     x    2        \  x            /   2*\3  - log(x)/
-- + 3 *log (3) - ------------------- + ---------------
 2                       -4 + x                    2   
x                                          (-4 + x)    
-------------------------------------------------------
                         -4 + x

$$\frac{3^{x} \log{\left(3 \right)}^{2} + \frac{2 \left(3^{x} - \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{2 \left(3^{x} \log{\left(3 \right)} - \frac{1}{x}\right)}{x - 4} + \frac{1}{x^{2}}}{x - 4}$$

Simplify

The third derivative [src]

                                        /1     x    2   \                      
                                      3*|-- + 3 *log (3)|     /  1    x       \
                      / x         \     | 2             |   6*|- - + 3 *log(3)|
  2     x    3      6*\3  - log(x)/     \x              /     \  x            /
- -- + 3 *log (3) - --------------- - ------------------- + -------------------
   3                           3             -4 + x                      2     
  x                    (-4 + x)                                  (-4 + x)      
-------------------------------------------------------------------------------
                                     -4 + x

$$\frac{3^{x} \log{\left(3 \right)}^{3} - \frac{6 \left(3^{x} - \log{\left(x \right)}\right)}{\left(x - 4\right)^{3}} - \frac{3 \left(3^{x} \log{\left(3 \right)}^{2} + \frac{1}{x^{2}}\right)}{x - 4} + \frac{6 \left(3^{x} \log{\left(3 \right)} - \frac{1}{x}\right)}{\left(x - 4\right)^{2}} - \frac{2}{x^{3}}}{x - 4}$$

Simplify

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Derivative of (3^x-ln(x))/(x-4)

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The solution