Derivative of ln(tgx+ctgx)/(sina)

The solution

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log(tan(x) + cot(x))
--------------------
       sin(a)

$$\frac{\log{\left(\tan{\left(x \right)} + \cot{\left(x \right)} \right)}}{\sin{\left(a \right)}}$$

log(tan(x) + cot(x))/sin(a)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \tan{\left(x \right)} + \cot{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)$ :
  1. Differentiate $\tan{\left(x \right)} + \cot{\left(x \right)}$ term by term:
    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:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    3. There are multiple ways to do this derivative.
      Method #1
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
      
      Let $u = \tan{\left(x \right)}$ .
      
      Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
      Method #2
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}}{\tan{\left(x \right)} + \cot{\left(x \right)}}$
So, the result is: $\frac{\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}}{\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) \sin{\left(a \right)}}$
Now simplify:

$\frac{\tan{\left(x \right)} - \frac{1}{\tan{\left(x \right)}}}{\sin{\left(a \right)}}$

The answer is:

$\frac{\tan{\left(x \right)} - \frac{1}{\tan{\left(x \right)}}}{\sin{\left(a \right)}}$

The first derivative [src]

      2         2       
   tan (x) - cot (x)    
------------------------
(tan(x) + cot(x))*sin(a)

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

Simplify

The second derivative [src]

                     2                                                  
  /   2         2   \                                                   
  \tan (x) - cot (x)/      /       2   \            /       2   \       
- -------------------- + 2*\1 + cot (x)/*cot(x) + 2*\1 + tan (x)/*tan(x)
    cot(x) + tan(x)                                                     
------------------------------------------------------------------------
                        (cot(x) + tan(x))*sin(a)

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

Simplify

The third derivative [src]

   /                                                     3                                                                                                                          \
   |             2                2   /   2         2   \                                                          /   2         2   \ //       2   \          /       2   \       \|
   |/       2   \    /       2   \    \tan (x) - cot (x)/         2    /       2   \        2    /       2   \   3*\tan (x) - cot (x)/*\\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/|
-2*|\1 + cot (x)/  - \1 + tan (x)/  - -------------------- - 2*tan (x)*\1 + tan (x)/ + 2*cot (x)*\1 + cot (x)/ + -------------------------------------------------------------------|
   |                                                    2                                                                                  cot(x) + tan(x)                          |
   \                                   (cot(x) + tan(x))                                                                                                                            /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               (cot(x) + tan(x))*sin(a)

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

Simplify

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Derivative of ln(tgx+ctgx)/(sina)

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The solution

Method #1

Method #2