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Integral of d{x}:
tg^2x-ctg^2x
Identical expressions
y=tg^2x-ctg^2x
y equally tg squared x minus ctg squared x
y=tg2x-ctg2x
y=tg²x-ctg²x
y=tg to the power of 2x-ctg to the power of 2x
Similar expressions
y=tg^2x+ctg^2x

The derivative of the function/ y=tg^2x-ctg^2x

Derivative of y=tg^2x-ctg^2x

The solution

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   2         2   
tan (x) - cot (x)

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

d /   2         2   \
--\tan (x) - cot (x)/
dx

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

Transform

Detail solution

Differentiate $\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}$ term by term:
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
  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 $\frac{d}{d x} f{\left(x \right)}$ :
    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 of the chain rule is:
  
  $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cot{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(x \right)}$ :
    1. There are multiple ways to do this derivative.
      Method #1
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
      
      Let $u = \tan{\left(x \right)}$ .
      
      Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
      Method #2
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    The result of the chain rule is:
    
    $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  So, the result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
Now simplify:

$\frac{2 \left(\tan^{3}{\left(x \right)} + \cot{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$

The answer is:

$\frac{2 \left(\tan^{3}{\left(x \right)} + \cot{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/         2   \          /          2   \       
\2 + 2*tan (x)/*tan(x) - \-2 - 2*cot (x)/*cot(x)

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

Simplify

The second derivative [src]

  /             2                2                                                    \
  |/       2   \    /       2   \         2    /       2   \        2    /       2   \|
2*\\1 + tan (x)/  - \1 + cot (x)/  - 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)//

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

Simplify

The third derivative [src]

  /                                                               2                         2       \
  |   3    /       2   \      3    /       2   \     /       2   \             /       2   \        |
8*\cot (x)*\1 + cot (x)/ + tan (x)*\1 + tan (x)/ + 2*\1 + cot (x)/ *cot(x) + 2*\1 + tan (x)/ *tan(x)/

$$8 \cdot \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{3}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of y=tg^2x-ctg^2x

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Piecewise:

The solution

Method #1

Method #2