Derivative of y=tgx/sinx-cosx

The solution

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tan(x)         
------ - cos(x)
sin(x)

$$- \cos{\left(x \right)} + \frac{\tan{\left(x \right)}}{\sin{\left(x \right)}}$$

tan(x)/sin(x) - cos(x)

Detail solution

Differentiate $- \cos{\left(x \right)} + \frac{\tan{\left(x \right)}}{\sin{\left(x \right)}}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \tan{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  So, the result is: $\sin{\left(x \right)}$
The result is: $\frac{\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

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The first derivative [src]

       2                            
1 + tan (x)   cos(x)*tan(x)         
----------- - ------------- + sin(x)
   sin(x)           2               
                 sin (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                         2                                                                                                                                           
            /       2   \      /       2   \        3                                    2    /       2   \        2    /       2   \     /       2   \              
          2*\1 + tan (x)/    3*\1 + tan (x)/   6*cos (x)*tan(x)   5*cos(x)*tan(x)   4*tan (x)*\1 + tan (x)/   6*cos (x)*\1 + tan (x)/   6*\1 + tan (x)/*cos(x)*tan(x)
-sin(x) + ---------------- + --------------- - ---------------- - --------------- + ----------------------- + ----------------------- - -----------------------------
               sin(x)             sin(x)              4                  2                   sin(x)                      3                            2              
                                                   sin (x)            sin (x)                                         sin (x)                      sin (x)

$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\sin{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{\sin{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{\sin{\left(x \right)}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \sin{\left(x \right)} - \frac{5 \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \cos^{3}{\left(x \right)} \tan{\left(x \right)}}{\sin^{4}{\left(x \right)}}$$

Simplify

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Derivative of y=tgx/sinx-cosx

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