Derivative of y=cosx/4tgx

The solution

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cos(x)       
------*tan(x)
  4

$$\frac{\cos{\left(x \right)}}{4} \tan{\left(x \right)}$$

(cos(x)/4)*tan(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  $g{\left(x \right)} = \tan{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $4$ is zero.
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=cosx/4tgx

The graph:

Piecewise:

The solution