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The derivative of the function/ e^(3x)*ln(3x)

Derivative of e^(3x)*ln(3x)

The solution

You have entered [src]

 3*x         
E   *log(3*x)

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

E^(3*x)*log(3*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = e^{3 x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  The result of the chain rule is:
  
  $3 e^{3 x}$
$g{\left(x \right)} = \log{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  The result of the chain rule is:
  
  $\frac{1}{x}$
The result is: $3 e^{3 x} \log{\left(3 x \right)} + \frac{e^{3 x}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 3*x                  
e         3*x         
---- + 3*e   *log(3*x)
 x

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

Simplify

The second derivative [src]

/  1    6             \  3*x
|- -- + - + 9*log(3*x)|*e   
|   2   x             |     
\  x                  /

$$\left(9 \log{\left(3 x \right)} + \frac{6}{x} - \frac{1}{x^{2}}\right) e^{3 x}$$

Simplify

The third derivative [src]

/  9    2    27              \  3*x
|- -- + -- + -- + 27*log(3*x)|*e   
|   2    3   x               |     
\  x    x                    /

$$\left(27 \log{\left(3 x \right)} + \frac{27}{x} - \frac{9}{x^{2}} + \frac{2}{x^{3}}\right) e^{3 x}$$

Simplify

The graph

Derivative of e^(3x)*ln(3x)