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Identical expressions
y=ln^ three *sinx/ three
y equally ln cubed multiply by sinus of x divide by 3
y equally ln to the power of three multiply by sinus of x divide by three
y=ln3*sinx/3
y=ln³*sinx/3
y=ln to the power of 3*sinx/3
y=ln^3sinx/3
y=ln3sinx/3
y=ln^3*sinx divide by 3

The derivative of the function/ y=ln^3*sinx/3

Derivative of y=ln^3*sinx/3

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

   3                2          
log (x)*cos(x)   log (x)*sin(x)
-------------- + --------------
      3                x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of y=ln^3*sinx/3

The graph:

Piecewise:

The solution