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The derivative of the function/ 3^(x)*(log(x))/(log3)

Derivative of 3^(x)*(log(x))/(log3)

Function f() - derivative -N order at the point
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The graph:

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Piecewise:

The solution

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

  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=3xf{\left(x \right)} = 3^{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

      The result is: 3xlog⁡(3)log⁡(x)+3xx3^{x} \log{\left(3 \right)} \log{\left(x \right)} + \frac{3^{x}}{x}

    So, the result is: 3xlog⁡(3)log⁡(x)+3xxlog⁡(3)\frac{3^{x} \log{\left(3 \right)} \log{\left(x \right)} + \frac{3^{x}}{x}}{\log{\left(3 \right)}}

  2. Now simplify:

    3xlog⁡(x)+3xxlog⁡(3)3^{x} \log{\left(x \right)} + \frac{3^{x}}{x \log{\left(3 \right)}}

The answer is:

3xlog⁡(x)+3xxlog⁡(3)3^{x} \log{\left(x \right)} + \frac{3^{x}}{x \log{\left(3 \right)}}

The graph
02468-8-6-4-2-1010-200000200000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 x                   
3     x              
-- + 3 *log(3)*log(x)
x                    
---------------------
        log(3)
3xlog⁡(3)log⁡(x)+3xxlog⁡(3)\frac{3^{x} \log{\left(3 \right)} \log{\left(x \right)} + \frac{3^{x}}{x}}{\log{\left(3 \right)}}
Simplify
The second derivative [src] 
 x /  1       2             2*log(3)\
3 *|- -- + log (3)*log(x) + --------|
   |   2                       x    |
   \  x                             /
-------------------------------------
                log(3)
3x(log⁡(3)2log⁡(x)+2log⁡(3)x−1x2)log⁡(3)\frac{3^{x} \left(\log{\left(3 \right)}^{2} \log{\left(x \right)} + \frac{2 \log{\left(3 \right)}}{x} - \frac{1}{x^{2}}\right)}{\log{\left(3 \right)}}
Simplify
The third derivative [src] 
   /                                      2   \
 x |2       3             3*log(3)   3*log (3)|
3 *|-- + log (3)*log(x) - -------- + ---------|
   | 3                        2          x    |
   \x                        x                /
-----------------------------------------------
                     log(3)
3x(log⁡(3)3log⁡(x)+3log⁡(3)2x−3log⁡(3)x2+2x3)log⁡(3)\frac{3^{x} \left(\log{\left(3 \right)}^{3} \log{\left(x \right)} + \frac{3 \log{\left(3 \right)}^{2}}{x} - \frac{3 \log{\left(3 \right)}}{x^{2}} + \frac{2}{x^{3}}\right)}{\log{\left(3 \right)}}
Simplify
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