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The derivative of the function/ (1-2cos^2xtgx)

Derivative of (1-2cos^2xtgx)

The solution

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         2          
1 - 2*cos (x)*tan(x)

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

1 - 2*cos(x)^2*tan(x)

Detail solution

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

$- 2 \cos{\left(2 x \right)}$

The answer is:

$- 2 \cos{\left(2 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2    /       2   \                         
- 2*cos (x)*\1 + tan (x)/ + 4*cos(x)*sin(x)*tan(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /               2                                                                                                                                                              \
  |  /       2   \     2           2    /       2   \        2    /       2   \                                 2       2    /       2   \     /       2   \                     |
4*\- \1 + tan (x)/ *cos (x) - 3*sin (x)*\1 + tan (x)/ + 3*cos (x)*\1 + tan (x)/ - 4*cos(x)*sin(x)*tan(x) - 2*cos (x)*tan (x)*\1 + tan (x)/ + 6*\1 + tan (x)/*cos(x)*sin(x)*tan(x)/

$$4 \left(- \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos^{2}{\left(x \right)} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} - 4 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}\right)$$

Simplify

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Derivative of (1-2cos^2xtgx)

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The solution