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

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

Function f() - derivative -N order at the point
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   2         2   
tan (x) - cot (x)
tan⁡2(x)−cot⁡2(x)\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)} 
d /   2         2   \
--\tan (x) - cot (x)/
dx
ddx(tan⁡2(x)−cot⁡2(x))\frac{d}{d x} \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)
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Detail solution

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

    1. Let u=tan⁡(x)u = \tan{\left(x \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

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

      1. Rewrite the function to be differentiated:

        tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \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⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

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

        1. The derivative of sine is cosine:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

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

        1. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Now plug in to the quotient rule:

        sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      The result of the chain rule is:

      2(sin⁡2(x)+cos⁡2(x))tan⁡(x)cos⁡2(x)\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⁡(x)u = \cot{\left(x \right)}.

      2. Apply the power rule: u2u^{2} goes to 2u2 u

      3. Then, apply the chain rule. Multiply by ddxcot⁡(x)\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⁡(x)=1tan⁡(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

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

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

            The result of the chain rule is:

            −sin⁡2(x)+cos⁡2(x)cos⁡2(x)tan⁡2(x)- \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⁡(x)=cos⁡(x)sin⁡(x)\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \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⁡(x)f{\left(x \right)} = \cos{\left(x \right)} and g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

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

            1. The derivative of cosine is negative sine:

              ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

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

            1. The derivative of sine is cosine:

              ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

            Now plug in to the quotient rule:

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

        The result of the chain rule is:

        −2(sin⁡2(x)+cos⁡2(x))cot⁡(x)cos⁡2(x)tan⁡2(x)- \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: 2(sin⁡2(x)+cos⁡2(x))cot⁡(x)cos⁡2(x)tan⁡2(x)\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: 2(sin⁡2(x)+cos⁡2(x))tan⁡(x)cos⁡2(x)+2(sin⁡2(x)+cos⁡2(x))cot⁡(x)cos⁡2(x)tan⁡2(x)\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)}}

  2. Now simplify:

    2(tan⁡3(x)+cot⁡(x))cos⁡2(x)tan⁡2(x)\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:

2(tan⁡3(x)+cot⁡(x))cos⁡2(x)tan⁡2(x)\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
02468-8-6-4-2-1010-50000005000000
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)
(2tan⁡2(x)+2)tan⁡(x)−(−2cot⁡2(x)−2)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(2(tan⁡2(x)+1)tan⁡2(x)−2(cot⁡2(x)+1)cot⁡2(x)+(tan⁡2(x)+1)2−(cot⁡2(x)+1)2)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⋅((tan⁡2(x)+1)tan⁡3(x)+(cot⁡2(x)+1)cot⁡3(x)+2(tan⁡2(x)+1)2tan⁡(x)+2(cot⁡2(x)+1)2cot⁡(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
Derivative of y=tg^2x-ctg^2x
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