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y=tg^ two (x)+ctgx^ two
y equally tg squared (x) plus ctgx squared
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Similar expressions
y=tg^2(x)-ctgx^2

The derivative of the function/ y=tg^2(x)+ctgx^2

Derivative of y=tg^2(x)+ctgx^2

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. Let $u = \cot{\left(x \right)}$ .
5. Apply the power rule: $u^{2}$ goes to $2 u$
6. 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
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
    2. Let $u = \tan{\left(x \right)}$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      
      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
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ 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:
  
  $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
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*cot (x)/*cot(x) + \2 + 2*tan (x)/*tan(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 + cot (x)/  + \1 + tan (x)/  + 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)//

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=tg^2(x)+ctgx^2

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The solution

Method #1

Method #2