Mister Exam

Other calculators

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

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /x\      /x\
tan|-| - cot|-|
   \2/      \2/
tan⁡(x2)−cot⁡(x2)\tan{\left(\frac{x}{2} \right)} - \cot{\left(\frac{x}{2} \right)} 
tan(x/2) - cot(x/2)
Detail solution

  1. Differentiate tan⁡(x2)−cot⁡(x2)\tan{\left(\frac{x}{2} \right)} - \cot{\left(\frac{x}{2} \right)} term by term:

    1. Rewrite the function to be differentiated:

      tan⁡(x2)=sin⁡(x2)cos⁡(x2)\tan{\left(\frac{x}{2} \right)} = \frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}

    2. Apply the quotient rule, which is:

      ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x2)f{\left(x \right)} = \sin{\left(\frac{x}{2} \right)} and g(x)=cos⁡(x2)g{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}.

      To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Let u=x2u = \frac{x}{2}.

      2. The derivative of sine is cosine:

        ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: xx goes to 11

          So, the result is: 12\frac{1}{2}

        The result of the chain rule is:

        cos⁡(x2)2\frac{\cos{\left(\frac{x}{2} \right)}}{2}

      To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Let u=x2u = \frac{x}{2}.

      2. The derivative of cosine is negative sine:

        dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: xx goes to 11

          So, the result is: 12\frac{1}{2}

        The result of the chain rule is:

        −sin⁡(x2)2- \frac{\sin{\left(\frac{x}{2} \right)}}{2}

      Now plug in to the quotient rule:

      sin⁡2(x2)2+cos⁡2(x2)2cos⁡2(x2)\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \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⁡(x2)=1tan⁡(x2)\cot{\left(\frac{x}{2} \right)} = \frac{1}{\tan{\left(\frac{x}{2} \right)}}

        2. Let u=tan⁡(x2)u = \tan{\left(\frac{x}{2} \right)}.

        3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

        4. Then, apply the chain rule. Multiply by ddxtan⁡(x2)\frac{d}{d x} \tan{\left(\frac{x}{2} \right)}:

          1. Let u=x2u = \frac{x}{2}.

          2. ddutan⁡(u)=1cos⁡2(u)\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}

          3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

            1. The derivative of a constant times a function is the constant times the derivative of the function.

              1. Apply the power rule: xx goes to 11

              So, the result is: 12\frac{1}{2}

            The result of the chain rule is:

            12cos⁡2(x2)\frac{1}{2 \cos^{2}{\left(\frac{x}{2} \right)}}

          The result of the chain rule is:

          −sin⁡2(x2)2+cos⁡2(x2)2cos⁡2(x2)tan⁡2(x2)- \frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}

        Method #2

        1. Rewrite the function to be differentiated:

          cot⁡(x2)=cos⁡(x2)sin⁡(x2)\cot{\left(\frac{x}{2} \right)} = \frac{\cos{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}}

        2. Apply the quotient rule, which is:

          ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=cos⁡(x2)f{\left(x \right)} = \cos{\left(\frac{x}{2} \right)} and g(x)=sin⁡(x2)g{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}.

          To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

          1. Let u=x2u = \frac{x}{2}.

          2. The derivative of cosine is negative sine:

            dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

          3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

            1. The derivative of a constant times a function is the constant times the derivative of the function.

              1. Apply the power rule: xx goes to 11

              So, the result is: 12\frac{1}{2}

            The result of the chain rule is:

            −sin⁡(x2)2- \frac{\sin{\left(\frac{x}{2} \right)}}{2}

          To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

          1. Let u=x2u = \frac{x}{2}.

          2. The derivative of sine is cosine:

            ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

          3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

            1. The derivative of a constant times a function is the constant times the derivative of the function.

              1. Apply the power rule: xx goes to 11

              So, the result is: 12\frac{1}{2}

            The result of the chain rule is:

            cos⁡(x2)2\frac{\cos{\left(\frac{x}{2} \right)}}{2}

          Now plug in to the quotient rule:

          −sin⁡2(x2)2−cos⁡2(x2)2sin⁡2(x2)\frac{- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\sin^{2}{\left(\frac{x}{2} \right)}}

      So, the result is: sin⁡2(x2)2+cos⁡2(x2)2cos⁡2(x2)tan⁡2(x2)\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}

    The result is: sin⁡2(x2)2+cos⁡2(x2)2cos⁡2(x2)+sin⁡2(x2)2+cos⁡2(x2)2cos⁡2(x2)tan⁡2(x2)\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)}} + \frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}

  2. Now simplify:

    41−cos⁡(2x)\frac{4}{1 - \cos{\left(2 x \right)}}

The answer is:

41−cos⁡(2x)\frac{4}{1 - \cos{\left(2 x \right)}}

The graph
02468-8-6-4-2-1010-2000020000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       2/x\      2/x\
    cot |-|   tan |-|
        \2/       \2/
1 + ------- + -------
       2         2
tan⁡2(x2)2+cot⁡2(x2)2+1\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + 1
Simplify
The second derivative [src] 
/       2/x\\    /x\   /       2/x\\    /x\
|1 + tan |-||*tan|-| - |1 + cot |-||*cot|-|
\        \2//    \2/   \        \2//    \2/
-------------------------------------------
                     2
(tan⁡2(x2)+1)tan⁡(x2)−(cot⁡2(x2)+1)cot⁡(x2)2\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} - \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{2}
Simplify
The third derivative [src] 
             2                2                                                    
/       2/x\\    /       2/x\\         2/x\ /       2/x\\        2/x\ /       2/x\\
|1 + cot |-||  + |1 + tan |-||  + 2*cot |-|*|1 + cot |-|| + 2*tan |-|*|1 + tan |-||
\        \2//    \        \2//          \2/ \        \2//         \2/ \        \2//
-----------------------------------------------------------------------------------
                                         4
(tan⁡2(x2)+1)2+2(tan⁡2(x2)+1)tan⁡2(x2)+(cot⁡2(x2)+1)2+2(cot⁡2(x2)+1)cot⁡2(x2)4\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)} + \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot^{2}{\left(\frac{x}{2} \right)}}{4}
Simplify
The graph
Derivative of tan(x/2)-cot(x/2)
    © Mister Exam – Calculator