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Identical expressions
tan(x)-cot(x+n/ two)
tangent of (x) minus cotangent of (x plus n divide by 2)
tangent of (x) minus cotangent of (x plus n divide by two)
tanx-cotx+n/2
tan(x)-cot(x+n divide by 2)
Similar expressions
tan(x)+cot(x+n/2)
tan(x)-cot(x-n/2)

The derivative of the function/ tan(x)-cot(x+n/2)

Derivative of tan(x)-cot(x+n/2)

The solution

You have entered [src]

            /    n\
tan(x) - cot|x + -|
            \    2/

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

tan(x) - cot(x + n/2)

Detail solution

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

$\frac{1}{\cos^{2}{\left(x \right)}} + \frac{1}{\sin^{2}{\left(\frac{n}{2} + x \right)}}$

The answer is:

$\frac{1}{\cos^{2}{\left(x \right)}} + \frac{1}{\sin^{2}{\left(\frac{n}{2} + x \right)}}$

The first derivative [src]

       2/    n\      2   
2 + cot |x + -| + tan (x)
        \    2/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of tan(x)-cot(x+n/2)

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Piecewise:

The solution

Method #1

Method #2