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Sum of series:
f(x)
Integral of d{x}:
f(x)
f(x)
Identical expressions
f(x)= three ^x*log3x
f(x) equally 3 to the power of x multiply by logarithm of 3x
f(x) equally three to the power of x multiply by logarithm of 3x
f(x)=3x*log3x
fx=3x*log3x
f(x)=3^xlog3x
f(x)=3xlog3x
fx=3xlog3x
fx=3^xlog3x

The derivative of the function/ f(x)=3^x*log3x

Derivative of f(x)=3^x*log3x

The solution

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 x         
3 *log(3*x)

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

3^x*log(3*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 3^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. $\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}$
$g{\left(x \right)} = \log{\left(3 x \right)}$ ; 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: $3^{x} \log{\left(3 \right)} \log{\left(3 x \right)} + \frac{3^{x}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of f(x)=3^x*log3x

The graph:

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The solution