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Identical expressions
three ^(x)*(log(x))/(log3)
3 to the power of (x) multiply by ( logarithm of (x)) divide by ( logarithm of 3)
three to the power of (x) multiply by ( logarithm of (x)) divide by ( logarithm of 3)
3(x)*(log(x))/(log3)
3x*logx/log3
3^(x)(log(x))/(log3)
3(x)(log(x))/(log3)
3xlogx/log3
3^xlogx/log3
3^(x)*(log(x)) divide by (log3)

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

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

The solution

You have entered [src]

 x       
3 *log(x)
---------
  log(3)

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

(3^x*log(x))/log(3)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 3^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. $\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $3^{x} \log{\left(3 \right)} \log{\left(x \right)} + \frac{3^{x}}{x}$
So, the result is: $\frac{3^{x} \log{\left(3 \right)} \log{\left(x \right)} + \frac{3^{x}}{x}}{\log{\left(3 \right)}}$
Now simplify:

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

The answer is:

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

The graph

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)

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

Simplify

The second derivative [src]

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

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

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)

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

Simplify

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Identical expressions

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

The graph:

Piecewise:

The solution