Derivative of tgx-ctgx+cosx

The solution

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tan(x) - cot(x) + cos(x)

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

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Detail solution

Differentiate $\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) + \cos{\left(x \right)}$ term by term:
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. The derivative of a constant times a function is the constant times the derivative of the function.
    1. 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:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    So, the result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of tgx-ctgx+cosx

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Piecewise:

The solution

Method #1

Method #2