Derivative of sin^5lnx/2

The solution

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   5          
sin (x)*log(x)
--------------
      2

$$\frac{\log{\left(x \right)} \sin^{5}{\left(x \right)}}{2}$$

(sin(x)^5*log(x))/2

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin^{5}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
  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:
    
    $5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $5 \log{\left(x \right)} \sin^{4}{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{5}{\left(x \right)}}{x}$
So, the result is: $\frac{5 \log{\left(x \right)} \sin^{4}{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\sin^{5}{\left(x \right)}}{2 x}$
Now simplify:

$\frac{\left(5 x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{4}{\left(x \right)}}{2 x}$

The answer is:

$\frac{\left(5 x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{4}{\left(x \right)}}{2 x}$

The graph

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Plot the graph f'(x)

The first derivative [src]

   5           4                 
sin (x)   5*sin (x)*cos(x)*log(x)
------- + -----------------------
  2*x                2

$$\frac{5 \log{\left(x \right)} \sin^{4}{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\sin^{5}{\left(x \right)}}{2 x}$$

Simplify

The second derivative [src]

         /   2                                                       \ 
    3    |sin (x)     /   2           2   \          10*cos(x)*sin(x)| 
-sin (x)*|------- + 5*\sin (x) - 4*cos (x)/*log(x) - ----------------| 
         |    2                                             x        | 
         \   x                                                       / 
-----------------------------------------------------------------------
                                   2

$$- \frac{\left(5 \left(\sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} - \frac{10 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin^{3}{\left(x \right)}}{2}$$

Simplify

The third derivative [src]

         /       3                                                       /   2           2   \                2          \ 
    2    |  2*sin (x)     /        2            2   \                 15*\sin (x) - 4*cos (x)/*sin(x)   15*sin (x)*cos(x)| 
-sin (x)*|- --------- + 5*\- 12*cos (x) + 13*sin (x)/*cos(x)*log(x) + ------------------------------- + -----------------| 
         |       3                                                                   x                           2       | 
         \      x                                                                                               x        / 
---------------------------------------------------------------------------------------------------------------------------
                                                             2

$$- \frac{\left(5 \left(13 \sin^{2}{\left(x \right)} - 12 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} \cos{\left(x \right)} + \frac{15 \left(\sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{x} + \frac{15 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{3}{\left(x \right)}}{x^{3}}\right) \sin^{2}{\left(x \right)}}{2}$$

Simplify

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Identical expressions

Derivative of sin^5lnx/2

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The solution