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Identical expressions
(3x*ln(sin(x)))/ four /3x
(3x multiply by ln( sinus of (x))) divide by 4 divide by 3x
(3x multiply by ln( sinus of (x))) divide by four divide by 3x
(3xln(sin(x)))/4/3x
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(3x*ln(sin(x))) divide by 4 divide by 3x
Similar expressions
(3x*ln(sinx))/4/3x

The derivative of the function/ (3x*ln(sin(x)))/4/3x

Derivative of (3x*ln(sin(x)))/4/3x

The solution

You have entered [src]

/3*x*log(sin(x))\  
|---------------|  
\       4       /  
-----------------*x
        3

$$x \frac{\frac{1}{4} \cdot 3 x \log{\left(\sin{\left(x \right)} \right)}}{3}$$

((((3*x)*log(sin(x)))/4)/3)*x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 3 x^{2} \log{\left(\sin{\left(x \right)} \right)}$ and $g{\left(x \right)} = 12$ .

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 product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    $g{\left(x \right)} = \log{\left(\sin{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \sin{\left(x \right)}$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      The result of the chain rule is:
      
      $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
    The result is: $\frac{x^{2} \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x \log{\left(\sin{\left(x \right)} \right)}$
  So, the result is: $\frac{3 x^{2} \cos{\left(x \right)}}{\sin{\left(x \right)}} + 6 x \log{\left(\sin{\left(x \right)} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $12$ is zero.
Now plug in to the quotient rule:

$\frac{x^{2} \cos{\left(x \right)}}{4 \sin{\left(x \right)}} + \frac{x \log{\left(\sin{\left(x \right)} \right)}}{2}$
Now simplify:

$\frac{x \left(\frac{x}{\tan{\left(x \right)}} + 2 \log{\left(\sin{\left(x \right)} \right)}\right)}{4}$

The answer is:

$\frac{x \left(\frac{x}{\tan{\left(x \right)}} + 2 \log{\left(\sin{\left(x \right)} \right)}\right)}{4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                             /3*x*log(sin(x))\
                             |---------------|
  /log(sin(x))   x*cos(x)\   \       4       /
x*|----------- + --------| + -----------------
  \     4        4*sin(x)/           3

$$x \left(\frac{x \cos{\left(x \right)}}{4 \sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{4}\right) + \frac{\frac{1}{4} \cdot 3 x \log{\left(\sin{\left(x \right)} \right)}}{3}$$

Simplify

The second derivative [src]

                  /  /       2   \           \             
                  |  |    cos (x)|   2*cos(x)|   2*x*cos(x)
2*log(sin(x)) - x*|x*|1 + -------| - --------| + ----------
                  |  |       2   |    sin(x) |     sin(x)  
                  \  \    sin (x)/           /             
-----------------------------------------------------------
                             4

$$\frac{- x \left(x \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) + \frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 \log{\left(\sin{\left(x \right)} \right)}}{4}$$

Simplify

The third derivative [src]

      /       2   \                /       2   \                  
      |    cos (x)|   6*cos(x)     |    cos (x)| /     2*x*cos(x)\
- 3*x*|1 + -------| + -------- + x*|1 + -------|*|-3 + ----------|
      |       2   |    sin(x)      |       2   | \       sin(x)  /
      \    sin (x)/                \    sin (x)/                  
------------------------------------------------------------------
                                4

$$\frac{x \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 3\right) - 3 x \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{4}$$

Simplify

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Derivative of (3x*ln(sin(x)))/4/3x

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The solution