Derivative of xsec^2x-tgx

The solution

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

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

d /     2            \
--\x*sec (x) - tan(x)/
dx

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

Transform

Detail solution

Differentiate $x \sec^{2}{\left(x \right)} - \tan{\left(x \right)}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sec^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $\frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{2 x \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \sec^{2}{\left(x \right)}$
2. 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: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{2 x \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \sec^{2}{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /               2                                                                                                                                 \
  |  /       2   \         2    /       2   \        2    /       2   \        2       2             2       3             2    /       2   \       |
2*\- \1 + tan (x)/  - 2*tan (x)*\1 + tan (x)/ + 3*sec (x)*\1 + tan (x)/ + 6*sec (x)*tan (x) + 4*x*sec (x)*tan (x) + 8*x*sec (x)*\1 + tan (x)/*tan(x)/

$$2 \cdot \left(4 x \tan^{3}{\left(x \right)} \sec^{2}{\left(x \right)} + 8 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec^{2}{\left(x \right)} + 6 \tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{2}{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right)^{2}\right)$$

Simplify

The graph

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Derivative of xsec^2x-tgx

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The solution