Derivative of (sinx+cosx)(tanx+cotx)

The solution

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

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

d                                      
--((sin(x) + cos(x))*(tan(x) + cot(x)))
dx

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(x \right)} + \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\sin{\left(x \right)} + \cos{\left(x \right)}$ term by term:
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $- \sin{\left(x \right)} + \cos{\left(x \right)}$
$g{\left(x \right)} = \tan{\left(x \right)} + \cot{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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
    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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
The result is: $\left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) + \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)$
Now simplify:

$\frac{1}{\cos{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}} + \frac{4 \sin{\left(x \right)}}{\cos{\left(2 x \right)} + 1} - \frac{8 \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{1 - \cos{\left(4 x \right)}}$

The answer is:

$\frac{1}{\cos{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}} + \frac{4 \sin{\left(x \right)}}{\cos{\left(2 x \right)} + 1} - \frac{8 \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{1 - \cos{\left(4 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (sinx+cosx)(tanx+cotx)

The graph:

Piecewise:

The solution

Method #1

Method #2