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Identical expressions
cos^4x*ln2/x
co sinus of e of to the power of 4x multiply by ln2 divide by x
cos4x*ln2/x
cos⁴x*ln2/x
cos^4xln2/x
cos4xln2/x
cos^4x*ln2 divide by x

The derivative of the function/ cos^4x*ln2/x

Derivative of cos^4x*ln2/x

The solution

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   4          
cos (x)*log(2)
--------------
      x

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

(cos(x)^4*log(2))/x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(2 \right)} \cos^{4}{\left(x \right)}$ and $g{\left(x \right)} = x$ .

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. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$
  So, the result is: $- 4 \log{\left(2 \right)} \sin{\left(x \right)} \cos^{3}{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$\frac{- 4 x \log{\left(2 \right)} \sin{\left(x \right)} \cos^{3}{\left(x \right)} - \log{\left(2 \right)} \cos^{4}{\left(x \right)}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     4                  3                 
  cos (x)*log(2)   4*cos (x)*log(2)*sin(x)
- -------------- - -----------------------
         2                    x           
        x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /     3                                             /     2           2   \                2          \              
   |3*cos (x)     /       2           2   \          6*\- cos (x) + 3*sin (x)/*cos(x)   12*cos (x)*sin(x)|              
-2*|--------- + 4*\- 5*cos (x) + 3*sin (x)/*sin(x) + -------------------------------- + -----------------|*cos(x)*log(2)
   |     3                                                          x                            2       |              
   \    x                                                                                       x        /              
------------------------------------------------------------------------------------------------------------------------
                                                           x

$$- \frac{2 \left(4 \left(3 \sin^{2}{\left(x \right)} - 5 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} + \frac{6 \left(3 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{x} + \frac{12 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{3 \cos^{3}{\left(x \right)}}{x^{3}}\right) \log{\left(2 \right)} \cos{\left(x \right)}}{x}$$

Simplify

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Derivative of cos^4x*ln2/x

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The solution