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tgx+ctgx
The derivative of the function/ tgx+ctgx

Derivative of tgx+ctgx

Function f() - derivative -N order at the point
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You have entered [src] 
tan(x) + cot(x)
tan⁡(x)+cot⁡(x)\tan{\left(x \right)} + \cot{\left(x \right)} 
d                  
--(tan(x) + cot(x))
dx
ddx(tan⁡(x)+cot⁡(x))\frac{d}{d x} \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)
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Detail solution

  1. Differentiate tan⁡(x)+cot⁡(x)\tan{\left(x \right)} + \cot{\left(x \right)} term by term:

    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)}}

    3. 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 is: sin⁡2(x)+cos⁡2(x)cos⁡2(x)−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)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

  2. Now simplify:

    1cos⁡2(x)−1sin⁡2(x)\frac{1}{\cos^{2}{\left(x \right)}} - \frac{1}{\sin^{2}{\left(x \right)}}

The answer is:

1cos⁡2(x)−1sin⁡2(x)\frac{1}{\cos^{2}{\left(x \right)}} - \frac{1}{\sin^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   2         2   
tan (x) - cot (x)
tan⁡2(x)−cot⁡2(x)\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}
Simplify
The second derivative [src] 
  //       2   \          /       2   \       \
2*\\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/
2((tan⁡2(x)+1)tan⁡(x)+(cot⁡2(x)+1)cot⁡(x))2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)
Simplify
The third 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 graph
Derivative of tgx+ctgx
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