Derivative of (secx+tanx)(secx-tanx)

The solution

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan{\left(x \right)} + \sec{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = - \tan{\left(x \right)} + \sec{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $- \tan{\left(x \right)} + \sec{\left(x \right)}$ term by term:
  1. The derivative of secant is secant times tangent:
    
    $\frac{d}{d x} \sec{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
    So, the result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  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 is: $\left(- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right) + \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \left(- \tan{\left(x \right)} + \sec{\left(x \right)}\right)$
Now simplify:

$0$

The answer is:

$0$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (secx+tanx)(secx-tanx)

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