Derivative of 3^x*logx

You have entered [src]

 x       
3 *log(x)

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

3^x*log(x)

Detail solution

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

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

The answer is:

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

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

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

Simplify

The second derivative [src]

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

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

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                /

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

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of 3^x*logx

The graph:

Piecewise:

The solution