Derivative of ln(secx+tanx)

The solution

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log(sec(x) + tan(x))

$$\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}$$

log(sec(x) + tan(x))

Detail solution

Let $u = \tan{\left(x \right)} + \sec{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right)$ :
1. Differentiate $\tan{\left(x \right)} + \sec{\left(x \right)}$ term by term:
  1. Rewrite the function to be differentiated:
    
    $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
  2. Let $u = \cos{\left(x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  5. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  6. 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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

$\frac{\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}}{\tan{\left(x \right)} + \sec{\left(x \right)}}$
Now simplify:

$\frac{1}{\cos{\left(x \right)}}$

The answer is:

$\frac{1}{\cos{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                   
1 + tan (x) + sec(x)*tan(x)
---------------------------
      sec(x) + tan(x)

$$\frac{\tan^{2}{\left(x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 1}{\tan{\left(x \right)} + \sec{\left(x \right)}}$$

Simplify

The second derivative [src]

                                                                     2                         
                                        /       2                   \                          
   2             /       2   \          \1 + tan (x) + sec(x)*tan(x)/      /       2   \       
tan (x)*sec(x) + \1 + tan (x)/*sec(x) - ------------------------------ + 2*\1 + tan (x)/*tan(x)
                                               sec(x) + tan(x)                                 
-----------------------------------------------------------------------------------------------
                                        sec(x) + tan(x)

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

Simplify

The third derivative [src]

                                                                   3                                                                                                                                                             
               2                      /       2                   \                                /       2                   \ /   2             /       2   \            /       2   \       \                                
  /       2   \       3             2*\1 + tan (x) + sec(x)*tan(x)/         2    /       2   \   3*\1 + tan (x) + sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/     /       2   \              
2*\1 + tan (x)/  + tan (x)*sec(x) + -------------------------------- + 4*tan (x)*\1 + tan (x)/ - ------------------------------------------------------------------------------------------------ + 5*\1 + tan (x)/*sec(x)*tan(x)
                                                            2                                                                            sec(x) + tan(x)                                                                         
                                           (sec(x) + tan(x))                                                                                                                                                                     
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                         sec(x) + tan(x)

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

Simplify

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Derivative of ln(secx+tanx)

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