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The derivative of the function/ (3^x-ln(x))/(x-4)

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

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
 x         
3  - log(x)
-----------
   x - 4
3x−log⁡(x)x−4\frac{3^{x} - \log{\left(x \right)}}{x - 4} 
(3^x - log(x))/(x - 4)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=3x−log⁡(x)f{\left(x \right)} = 3^{x} - \log{\left(x \right)} and g(x)=x−4g{\left(x \right)} = x - 4.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 3x−log⁡(x)3^{x} - \log{\left(x \right)} term by term:

      1. ddx3x=3xlog⁡(3)\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⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

        So, the result is: −1x- \frac{1}{x}

      The result is: 3xlog⁡(3)−1x3^{x} \log{\left(3 \right)} - \frac{1}{x}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x−4x - 4 term by term:

      1. The derivative of the constant −4-4 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

    −3x+(x−4)(3xlog⁡(3)−1x)+log⁡(x)(x−4)2\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}}

  2. Now simplify:

    x(−3x+log⁡(x))+(x−4)(3xxlog⁡(3)−1)x(x−4)2\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:

x(−3x+log⁡(x))+(x−4)(3xxlog⁡(3)−1)x(x−4)2\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
02468-8-6-4-2-1010-5000050000
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)
−3x−log⁡(x)(x−4)2+3xlog⁡(3)−1xx−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
3xlog⁡(3)2+2(3x−log⁡(x))(x−4)2−2(3xlog⁡(3)−1x)x−4+1x2x−4\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
3xlog⁡(3)3−6(3x−log⁡(x))(x−4)3−3(3xlog⁡(3)2+1x2)x−4+6(3xlog⁡(3)−1x)(x−4)2−2x3x−4\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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