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secx^ two -2tanx
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Similar expressions
secx^2+2tanx

The derivative of the function/ secx^2-2tanx

Derivative of secx^2-2tanx

The solution

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

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

sec(x)^2 - 2*tan(x)

Detail solution

Differentiate $- 2 \tan{\left(x \right)} + \sec^{2}{\left(x \right)}$ term by term:
1. Let $u = \sec{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sec{\left(x \right)}$ :
  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:
    
    $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
The result is: $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of secx^2-2tanx

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Piecewise:

The solution