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Similar expressions
3*x*ln(3*x)*sinx

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

Derivative of 3xln(3x)sin(x)

The solution

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3*x*log(3*x)*sin(x)

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

((3*x)*log(3*x))*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 3 x \log{\left(3 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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. 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: $3$
  $g{\left(x \right)} = \log{\left(3 x \right)}$ ; 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: $3 \log{\left(3 x \right)} + 3$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result is: $3 x \log{\left(3 x \right)} \cos{\left(x \right)} + \left(3 \log{\left(3 x \right)} + 3\right) \sin{\left(x \right)}$
Now simplify:

$3 x \log{\left(3 x \right)} \cos{\left(x \right)} + 3 \left(\log{\left(3 x \right)} + 1\right) \sin{\left(x \right)}$

The answer is:

$3 x \log{\left(3 x \right)} \cos{\left(x \right)} + 3 \left(\log{\left(3 x \right)} + 1\right) \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

(3 + 3*log(3*x))*sin(x) + 3*x*cos(x)*log(3*x)

$$3 x \log{\left(3 x \right)} \cos{\left(x \right)} + \left(3 \log{\left(3 x \right)} + 3\right) \sin{\left(x \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Similar expressions

Derivative of 3xln(3x)sin(x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 3*x*ln(3*x)*sin(x)

The graph:

Piecewise:

The solution

Derivative of 3xln(3x)sin(x)