Derivative of cosx×tgx+sinx×ctgx

The solution

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cos(x)*tan(x) + sin(x)*cot(x)

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

cos(x)*tan(x) + sin(x)*cot(x)

Detail solution

Differentiate $\sin{\left(x \right)} \cot{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}$ term by term:
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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan{\left(x \right)}$
2. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  $g{\left(x \right)} = \cot{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. There are multiple ways to do this derivative.
    Method #1
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
    2. Let $u = \tan{\left(x \right)}$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. 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
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ 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: $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \cos{\left(x \right)} \cot{\left(x \right)}$
The result is: $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)} \cot{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of cosx×tgx+sinx×ctgx

The graph:

Piecewise:

The solution

Method #1

Method #2